MQ Gernal Maths HSC Course 2e SE

MATHS Quest General Mathematics HSC COURSE SE

ITION D E D CON

Robert Rowland

Second edition published 2007 by John Wiley & Sons Australia, Ltd 42 McDougall Street, Milton, Qld 4064 First edition published 2001 Typeset in 10.5/12.5 pt Times ©

John Wiley & Sons Australia, Ltd 2001, 2007

The moral rights of the author have been asserted. National Library of Australia Cataloguing-in-Publication data Rowland, Robert, 1963–. Maths quest: general mathematics HSC. 2nd ed. Includes index. For secondary school students. ISBN 978 0 7314 0569 5. (Student edition) ISBN 978 0 7314 0568 8. (Teacher edition) 1. Mathematics - Problems, exercises, etc. 2. Mathematics. I. Title. 510

Reproduction and communication for educational purposes The Australian Copyright Act 1968 allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. Cover photograph and internal design images: © Digital Vision Cartography by MAPgraphics Pty Ltd, Brisbane Illustrated by the Wiley Art Studio Printed in China by Printplus Limited 10 9 8 7 6 5 4 3

Contents Introduction vi About eBookPlus viii Acknowledgements ix

Summary 73 Chapter review 75 Practice examination questions

CHAPTER 3

CHAPTER 1 Credit and borrowing

1

Are you ready?

2 Flat rate interest 3 Exercise 1A 6 Home loans 9 Exercise 1B 12 10 Quick Questions 1 16 The cost of a loan 16 Exercise 1C 19 Investigation — Researching home loans 22 Credit cards 22 Exercise 1D 26 Investigation — Researching credit cards 28 10 Quick Questions 2 28 Loan repayments 29 Exercise 1E 31 Summary 34 Chapter review 36 Practice examination questions 39

CHAPTER 2 Further applications of area and volume 41 Are you ready?

78

42 Area of parts of the circle 43 Exercise 2A 45 Area of composite shapes 48 Exercise 2B 50 10 Quick Questions 1 52 Simpson’s rule 53 Exercise 2C 55 Surface area of cylinders and spheres 57 Exercise 2D 59 Investigation — Packaging 62 Volume of composite solids 62 Exercise 2E 65 Investigation — Maximising volume 67 10 Quick Questions 2 68 Error in measurement 69 Exercise 2F 71

Applications of trigonometry 79 Are you ready?

80 Review of right-angled triangles 81 Exercise 3A 85 Bearings 86 Exercise 3B 89 Investigation — Trigonometric ratios for obtuse angles 91 The sine rule 91 Investigation — Derivation of the sine rule 92 Exercise 3C 95 Exercise 3D 99 10 Quick Questions 1 101 Area of a triangle 102 Exercise 3E 103 The cosine rule 106 Investigation — Derivation of the cosine rule 106 Exercise 3F 109 Exercise 3G 115 10 Quick Questions 2 117 Radial surveys 118 Exercise 3H 121 Investigation — Conducting a radial survey 122 Summary 123 Chapter review 125 Practice examination questions 127

CHAPTER 4 Interpreting sets of data Are you ready?

129

130

Measures of location and spread 131 Exercise 4A 136 Skewness 140 Exercise 4B 142 10 Quick Questions 1 145 Displaying multiple data sets 145 Investigation — Examining exam results 146 Exercise 4C 150

iv Comparison of data sets 152 Exercise 4D 155 Investigation — Developing a two-way table 159 Summary 160 Chapter review 161 Practice examination questions 165

CHAPTER 5 Algebraic skills and techniques 167 Are you ready?

168

Substitution 169 Exercise 5A 171 Algebraic manipulation 172 Exercise 5B 174 10 Quick Questions 1 175 Equations and formulas 175 Exercise 5C 178 Solution by substitution 180 Exercise 5D 182 Investigation — Repeated enlargements 183 10 Quick Questions 2 183 Scientific notation 184 Exercise 5E 186 Summary 187 Chapter review 188 Practice examination questions 189

CHAPTER 6 Multi-stage events Are you ready?

191

192

Tree diagrams 193 Exercise 6A 195 Counting techniques 196 Investigation — Ordered arrangements 196 Investigation — Tree diagrams and ordered arrangements 198 Investigation — Committee selections 199 Investigation — Unordered selection 200 Exercise 6B 200 Probability and counting techniques 201 Investigation — Popular gaming 203 Exercise 6C 203 10 Quick Questions 1 204 Probability trees 205 Exercise 6D 209 Summary 212 Chapter review 213 Practice examination questions 215

CHAPTER 7 Applications of probability 217 Are you ready?

218

Expected outcomes 219 Investigation — Rolling a die 219 Exercise 7A 221 Financial expectation 223 Exercise 7B 225 10 Quick Questions 1 226 Two-way tables 227 Exercise 7C 229 Summary 232 Chapter review 233 Practice examination questions 235

CHAPTER 8 Annuities and loan repayments 237 Are you ready?

238

Future value of an annuity 239 Exercise 8A 242 10 Quick Questions 1 245 Present value of an annuity 246 Exercise 8B 248 Future and present value tables 250 Exercise 8C 253 10 Quick Questions 2 254 Loan repayments 255 Exercise 8D 257 Investigation — Types of loan arrangements 259 Summary 260 Chapter review 261 Practice examination questions 263

CHAPTER 9 Modelling linear and non-linear relationships Are you ready?

265

266

Linear functions 267 Exercise 9A 272 Investigation — Conversion of temperature 274 Quadratic functions 274 Exercise 9B 278 Investigation — Maximising areas 280 10 Quick Questions 1 280

v Other functions 281 Exercise 9C 284 Investigation — Compound interest 285 Variations 285 Exercise 9D 288 Graphing physical phenonema 289 Exercise 9E 292 Investigation — Force of gravity 293 Summary 294 Chapter review 295 Practice examination questions 297

CHAPTER 10 Depreciation Are you ready?

299

300

Modelling depreciation 301 Investigation — Depreciation of motor vehicles 301 Exercise 10A 304 Straight line depreciation 307 Exercise 10B 309 Declining balance method of depreciation 310 Exercise 10C 312 Investigation — Rates of depreciation 313 10 Quick Questions 1 314 Depreciation tables 314 Exercise 10D 319 Summary 322 Chapter review 323 Practice examination questions 325

CHAPTER 11 The normal distribution Are you ready?

327

328

z-scores 329 Exercise 11A 332 Comparison of scores 334 Exercise 11B 336 10 Quick Questions 1 338 Investigation — Comparison of subjects 338 Distribution of scores 339 Exercise 11C 342 Investigation — Examining a normal distribution 343 Summary 344 Chapter review 345 Practice examination questions 346

CHAPTER 12 Correlation Are you ready?

349

350

Scatterplots 351 Exercise 12A 355 Investigation — Collecting bivariate data 357 Regression lines 357 Exercise 12B 359 Exercise 12C 365 Investigation — Relationship between variables 369 10 Quick Questions 1 369 Correlation 370 Investigation — Causality 372 Exercise 12D 374 Summary 378 Chapter review 379 Practice examination questions 381

CHAPTER 13 Spherical geometry Are you ready?

383

384

Arc lengths 385 Exercise 13A 386 Great circles and small circles 389 Exercise 13B 390 10 Quick Questions 1 392 Latitude and longitude 393 Exercise 13C 396 Investigation — Important parallels of latitude 397 Distances on the Earth’s surface 397 Exercise 13D 399 10 Quick Questions 2 401 Time zones 401 Investigation — Australian time zones 402 Exercise 13E 404 Investigation — The keepers of time 405 Summary 406 Chapter review 407 Practice examination questions 409 Glossary 411 Formula sheet 414 Answers 417 Index 449

Introduction Maths Quest General Mathematics — HSC course is the second book in a series specifically designed for the General Mathematics Stage 6 Syllabus starting in 2000. This course replaces the current syllabuses for Mathematics in Society (1981) and Mathematics in Practice (1989). There are five new areas of study: • Financial mathematics • Data analysis • Measurement • Probability • Algebraic modelling. This resource contains: • a student textbook with accompanying eBookPLUS and • a teacher edition with accompanying eGuidePLUS.

Student textbook Full colour is used throughout to produce clearer graphs and diagrams, to provide bright, stimulating photos and to make navigation through the text easier. Clear, concise theory sections contain worked examples, highlighted important text and remember boxes. Worked examples in a Think/Write format provide a clear explanation of key steps and suggest a presentation for solutions. Exercises contain many carefully graded skills and application problems, including multiple-choice questions. Cross-references to relevant worked examples appear beside the first ‘matching’ question throughout the exercises. Investigations, including spreadsheet investigations, provide further learning opportunities through discovery. Sets of 10 Quick Questions allow students to quickly review the concepts just learnt before proceeding further in the chapter. A glossary of mathematical terms is provided to assist students’ understanding of the terminology introduced in each unit of the course. Words in bold type in the theory sections of each chapter are defined in the glossary at the back of the book. Each chapter concludes with a summary and chapter review exercise, containing questions in a variety of forms (multiple-choice, short-answer and analysis) that help consolidate students’ learning of new concepts. Practice examination questions provide a ready source of problems for students to use to gain further confidence in each topic.

vii Technology is fully integrated, in line with Board of Studies recommendations. As well as graphics calculators, Maths Quest features spreadsheets, dynamic geometry software and several graphing packages. Not only does the text promote these technologies as learning tools, but demonstration versions of the programs (with the exception of Microsoft Excel) are also included, as well as hundreds of supporting files available online. Graphics calculator tips are incorporated throughout the text. All formulas, which are given on the examination formula sheet, are marked with the symbol .

Programs included Graphmatica: an excellent graphing utility Equation grapher and regression analyser: like a graphics calculator for the PC GrafEq: graphs any relation, including complicated inequalities Poly: for visualising 3D polyhedra and their nets Tess: for producing tessellations and other symmetric planar illustrations TI Connect: calculator screen capture and program transfer CASIO Software FA-123: calculator screen capture and program transfer Cabri Geometry II: dynamic geometry program Adobe® Acrobat® Reader 4.0

Teacher edition with accompanying eGuidePLUS The teacher edition textbook contains everything in the student textbook and more. To support teachers assisting students in class, answers appear in red next to most questions in the exercises. Each exercise is annotated with relevant study design dot points. A readily accessible Work program lists all available resources and provides curriculum coverage information. The accompanying online resources contain everything in the student eBookPLUS and more. Four tests per chapter, fully worked solutions to WorkSHEETs, the work program and other curriculum advice in editable Word 2000 format are provided. Maths Quest is a rich collection of teaching and learning resources within one package. Maths Quest General Mathematics HSC course provides ample material, such as exercises, analysis questions, investigations, worksheets and technology files, from which teachers may set assessment tasks.

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Acknowledgements The Maths Quest project began in 1997, and the first edition of this book was printed in 2001. In that time we believe that Maths Quest has become the best resourced mathematical database in Australian education. I would like to thank all of those people who have supported us with our first edition. I hope that we have been able to help you in achieving your goals and have also played a part in your successes. Technology has evolved greatly since our first edition was published. The second edition has evolved from being a textbook in its first edition, into an interactive resource for both students and teachers. I would like to thank everyone at John Wiley & Sons Australia, Ltd for giving me the opportunity to do this. There are three people in particular that I would like to single out for special mention: Jennifer Nolan, whose support for the Maths Quest project and for me personally has made everything possible; Ingrid Kemp, the newest addition to our team. Ingrid has brought a new set of eyes to our project and kept the ball rolling — thanks, Ingrid; and finally, Keith Hartmann, who has tirelessly reviewed all of the new material and has completed all of the answer checking. Thanks, Keith — hope you’re enjoying retirement! Finally and most importantly to my family — thank you. Without your support this book and online resources would never have been completed. The author and publisher would like to thank the following copyright holders, organisations and individuals for their assistance and for permission to reproduce copyright material in this book.

Illustrative material • © Banana Stock: p. 302 • © Brand X Pictures: p. 134 • © Comstock: p. 143 • © Corbis Corporation: p. 67, p. 72, p. 77, p. 100, p. 126, p. 377 • © Digital Stock: p. 20/Corbis Corporation, p. 37/Corbis Corporation, p. 41/ Corbis Corporation, p. 58/Corbis Corporation, p. 81/Corbis Corporation, p. 167/Corbis Corporation, p. 179/Corbis Corporation, p. 249, Corbis Corporation, p. 304/Corbis Corporation, p. 327/Corbis Corporation, p. 332/Corbis Corporation, p. 397/Corbis Corporation, p. 408/Corbis Corporation • Digital Vision: p. 29, p. 144 (lower), p. 163, p. 231, p. 293, p. 383, p. 388, p. 392 • © Image Addict: p. 12 • © IT StockFree: p. 66, p. 198 • © John Wiley & Sons Australia: p. xi/Photo by Jo Patterson, p. 193/Photo by Werner Langer, p. 299/Taken by Kari-Ann Tapp • © MAPgraphics: p. 394/Map by MAPgraphics Pty Ltd, Brisbane. • © Photodisc: p. 1, p. 7, p. 8, p. 17, p. 32, p. 47, p. 53 (all), p. 55, p. 69, p. 79, p. 105, p. 116, p. 118, p. 125, p. 129, p. 139, p. 144 (upper), p. 158, p. 159, p. 172, p. 191, p. 195, p. 196, p. 205, p. 208, p. 210, p. 211, p. 213, p. 215, p. 217, p. 219, p. 221, p. 225, p. 234, p. 237, p. 239, p. 263, p. 265, p. 279, p. 289, p. 307, p. 312, p. 323, p. 337, p. 343, p. 346, p. 347, p. 349, p. 351, p. 356, p. 357 (both), p. 368, p. 370, p. 372, p. 380, p. 396 • © Purestock: p. 305 • © Stockbyte: p. 90

x Software The authors and publisher would like to thank the following software providers for their assistance and for permission to use their materials. However, the use of such material does not imply that the providers endorse this product in any way. Third party software — registered full version ordering information Full versions of third party software may be obtained by contacting the companies listed below. Texas Instruments TI Connect™ and TI-GRAPHLINK software TI Connect™ and TI-GRAPHLINK software reproduced with permission of the publisher Texas Instruments Incorporated. TI Connect software available from Texas Instruments Web: http://education.ti.com/us/product/software.html Note: The TI Connectivity cable can be purchased from educational booksellers or calculator suppliers. Casio FA-124 Software used with permission of Casio Computer Co. Ltd. © 2001 All rights reserved. Distributed by Shriro Australia Pty Ltd 23–27 Chaplin Drive Lane Cove NSW 2066 Web: www.casioed.net.au and find the calculator product range If you are interested in this product after expiry, please contact Shriro Australia Pty Ltd. Graphmatica Reproduced with permission of kSoft, Inc. 345 Montecillo Dr., Walnut Creek, CA 94595-2654. e-mail: [email protected] Web: http://www.graphmatica.com Software included is for evaluation purposes only. The user is expected to register share-ware if use exceeds 30 days. Order forms are available at www.graphmatica.com/register. txt Cabri Geometry™ II PLUS Reproduced with permission of Cabrilog. Cabrilog 6, Robert Schuman Place 38000 Grenoble FRANCE Web: http://www.cabri.com 1. Due to copyright restrictions, the demo version of Cabri Geometry™ II Plus must not be used in the classroom for presentation on a regular basis. 2. For site licences contact Cabrilog — Grenoble-France at «[email protected]» or www.cabri.com

xi GrafEq and Poly Evaluation copies of GrafEq™ and Poly™ have been included with permission from Pedagoguery Software, Inc. e-mail: [email protected] Web: http://www.peda.com Microsoft® Excel, Microsoft® Word and Microsoft® PowerPoint Microsoft Excel, Microsoft Word and Microsoft PowerPoint are registered trademarks of the Microsoft Corporation in the United States and/or other countries. Screenshots reproduced throughout with permission from Microsoft. Every effort has been made to trace the ownership of copyright material. Information that will enable the publisher to trace the copyright holders or to rectify any error or omission in subsequent reprints will be welcome. In such cases, please contact the Permission Section of John Wiley & Sons Australia, who will arrange for the payment of the usual fee.

About the author Robert Rowland has been teaching Mathematics for over 20 years and currently holds the position of Head teacher, Teaching and learning at Ulladulla High School. He taught at Cabramatta High School from 1985 to 1988 before taking up his appointment at Ulladulla High School in 1989. Robert has successfully taught all levels of Mathematics to Year 12 as well as Computing Studies 7–12 and Information Processes and Technology. Robert is the coauthor of New South Wales Maths Year 9 Standard and New South Wales Maths Year 10 Standard as well as being the author of Maths Quest General Mathematics — Preliminary Course and Maths Quest General Mathematics — HSC Course.

Credit and borrowing

1 syllabus reference Financial mathematics 4 • Credit and borrowing

In this chapter 1A 1B 1C 1D 1E

Flat rate interest Home loans The cost of a loan Credit cards Loan repayments

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

1.1

Converting a percentage to a decimal

1 Convert each of the following percentages to decimals. a 40% b 12% c 8% 1 e 0.3% f 7 --- % g 1--- % 2

1.2

4

Finding a percentage of a quantity (money)

2 Find: a 30% of $5000 d 0.45% of $3600

1.3

d 2.4% h 0.02%

b 5% of $7390 e 1--- % of $82 000 2

c 7.4% of $125 000 f 0.06% of $78 000

Calculating simple interest

3 Calculate the simple interest earned on an investment of: a $7000 at 9% p.a. for 4 years b $57 500 at 6.5% p.a. for 2 years c $90 000 at 7 1--- % for 2 1--- years d $60 000 at 5.2% p.a. for 9 months 2

1.4

2

Finding values of n and r in financial formulas

4 Find the value of n and r for each of the following investments a Interest of 6% p.a. for 5 years with interest calculated annually b Interest of 9% p.a. for 4 years with interest calculated six-monthly c Interest of 8.8% p.a. for 3 years with interest calculated quarterly d Interest of 7.2% p.a. for 10 years with interest calculated monthly e Interest of 21% p.a. for June with interest calculated daily

1.5

Calculating compound interest

5 Use the formula A = P(1 + r)n to calculate the amount to which each of the following investments will grow. a $7000 at 9% p.a. for 4 years with interest compounded annually b $75 000 at 6.2% p.a. for 6 years with interest compounded six-monthly Calculate the amount of compound interest earned on an investment of: c $18 000 at 9.2% p.a. for 3 years with interest compounded annually d $150 000 at 8.4% p.a. for 10 years with interest compounded quarterly

1.6

Substitution into a formula

6 Evaluate each of the following by substituting into the given formula. m a If d = ---- , find d when m = 30 and v = 3. v b If A = 1--- (x + y)h, find A when h = 10, x = 7 and y = 2. 2

c If s = ut + 1--- at2, find s when u = 0.8, t = 5 and a = 2.3. 2

Chapter 1 Credit and borrowing

3

Flat rate interest During the preliminary course we calculated the simple interest earned on investments. Flat rate interest is the borrowing equivalent of simple interest. Flat rate interest applies to many small loans and hire purchase agreements. When money is borrowed from a lending institution such as a bank at a flat rate of interest, the total amount of interest is calculated as a percentage of the initial amount borrowed and then this is multiplied by the term of the loan. The term of the loan is the length of time which the loan is agreed to be repaid over. The formula for calculating the amount of flat interest to be paid on a loan is the same formula as for simple interest (I): I = Prn where P = initial quantity r = percentage interest rate per period expressed as a decimal n = number of periods As you work through the financial mathematics strand there are several formulas that use the same pronumerals. While the initial quantity (P) will be the principal in an investing scenario, it will represent the amount borrowed in a loans situation. All of these formulas use the same pronumerals and all of them require r to be expressed as a decimal. It should be part of your normal practice when doing such questions to convert the interest rate, expressed as a percentage, to a decimal. In simple or flat rate interest, r will always be a rate per annum or per year and there will be no variation on this regardless of how often interest is paid. Similarly, n will always be the number of years of the investment or loan.

WORKED Example 1

Calculate the flat interest to be paid on a loan of $20 000 at 7.5% p.a. flat interest if the loan is to be repaid over 5 years. THINK

WRITE

1

Convert the interest rate to a decimal.

r = 7.5 ÷ 100 = 0.075

2

Write the formula.

I = Prn

3

Substitute the values of P, r (as a decimal) and n.

= $20 000

4

Calculate.

= $7500

0.075

5

Once the interest has been calculated, we can calculate the total amount that must be repaid in a loan. This is calculated by adding the principal and the interest.

4

Maths Quest General Mathematics HSC Course

WORKED Example 2

Alvin borrows $8000 to buy a car at a flat rate of 9% p.a. interest. Alvin is to repay the loan, plus interest, over 4 years. Calculate the total amount that Alvin is to repay on this loan. THINK

WRITE

1

Convert the interest rate to a decimal.

2

Write the interest formula. Substitute the values of P, r and n. Calculate the interest. Calculate the total repayments by adding the interest and principal.

3 4 5

r = 9 ÷ 100 = 0.09 I = Prn = $8000 0.09 4 = $2880 Total repayments = $8000 + $2880 Total repayments = $10 880

Graphics Calculator tip! Calculate simple interest Your Casio graphics calculator can perform a number of financial functions by using the TVM mode. One of the options in this mode is to calculate simple interest. Examples such as worked example 2 above are more simply done using the arithmetic method as shown above. However, for some of the more complex questions later in this chapter it is worth familiarising yourself with this method. 1. From the MENU select TVM.

2. Press F1 to select Simple Interest.

3. The calculator has two modes of calculating interest: 360 day mode or 365 day mode. You need to make sure that it is on 365 day mode. If not, press SHIFT [SET UP], select Date Mode and press F1 for 365. 4. Press EXE to return to the previous screen and enter the data for worked example 2. n = 4 365 (as n is in days) I% = 9 PV = –8000 (principal or present value is entered as a negative)

Chapter 1 Credit and borrowing

5

5. The calculator gives you two options. Press F1 (SI) for simple interest Press F2 (SFV) for future value (in other words, the principal plus interest). In this example we want the total repayments, so we press F2 (SPV). Most loans are repaid on a monthly basis. Once the total amount to be repaid has been calculated, this can be divided into equal monthly, fortnightly or weekly instalments.

WORKED Example 3 Narelle buys a computer on hire purchase. The cash price of the computer is $3000, but Narelle must pay a 10% deposit with the balance paid at 8% p.a. flat rate interest in equal monthly instalments over 3 years. a Calculate the deposit. b Calculate the balance owing. c Calculate the interest on the loan. d Calculate the total amount to be repaid. e Calculate the amount of each monthly instalment. THINK

WRITE

a Find 10% of $3000.

a Deposit = 10% of $3000 Deposit = $300

b Subtract the deposit from the cash price to find the amount borrowed.

b Balance = $3000 Balance = $2700

c

c I = Prn

1

Write the interest formula.

2

Substitute for P, r and n.

I = $2700

3

Calculate the interest.

I = $648

0.08

$300

3

d Add the interest to the amount borrowed.

d Total repayments = $2700 + $648 Total repayments = $3348

e Divide the total repayments by 36 (the number of monthly instalments in 3 years).

e Monthly repayments = $3348 ÷ 36 Monthly repayments = $93.00

If given the amount to be repaid each month, we can calculate the interest rate. The interest on the loan is the difference between the total repaid and the amount borrowed. This is then calculated as a yearly amount and written as a percentage of the amount borrowed.

6

Maths Quest General Mathematics HSC Course

WORKED Example 4

Theresa borrows $12 000 to buy a car. This is to be repaid over 5 years at $320 per month. Calculate the flat rate of interest that Theresa has been charged. THINK 1 2 3 4

WRITE

Calculate the total amount that is repaid. Subtract the principal from the total repayments to find the interest. Calculate the interest paid each year. Write the annual interest as a percentage of the amount borrowed.

Total repayments = $320 60 Total repayments = $19 200 Interest = $19 200 $12 000 Interest = $7200 Interest per year = $7200 ÷ 5 Interest per year = $1440 $1440 Interest rate = ------------------- 100% $12 000 Interest rate = 12%

remember 1. Flat rate interest is the borrowing equivalent of simple interest. It is calculated based on the initial amount borrowed. 2. The simple interest formula is used to calculate the amount of flat rate interest to be paid on a loan. The simple interest formula is I = Prn . 3. The total amount to be repaid on a loan is the principal plus interest. To calculate the amount of each instalment, we divide the total amount by the number of repayments. 4. When given the amount of each instalment, we can calculate the flat rate of interest.

SkillS

1A HEET

1.1 WORKED

SkillS

Converting Example a 1 percentage to a decimal HEET

1.2

SkillS

Finding a percentage of a WORKED quantity Example 2 HEET

Flat rate interest

1.3

Calculating simple interest

1 Calculate the amount of flat rate interest paid on each of the following loans. a $5000 at 7% p.a. for 2 years b $8000 at 5% p.a. for 3 years c $15 000 at 10% p.a. for 5 years d $9500 at 7.5% p.a. for 4 years e $2500 at 10.4% p.a. for 18 months 2 Roula buys a used car that has a cash price of $7500. She has saved a deposit of $2000 and borrows the balance at 9.6% p.a. flat rate to be repaid over 3 years. Calculate the amount of interest that Roula must pay. 3 Ben borrows $4000 for a holiday. The loan is to be repaid over 2 years at 12.5% p.a. flat interest. Calculate the total repayments that Ben must make. 4 Calculate the total amount to be paid on each of the following flat rate interest loans. a $3500 at 8% p.a. over 2 years b $13 500 at 11.6% p.a. over 5 years c $1500 at 13.5% p.a. over 18 months d $300 at 33% p.a. over 1 month e $100 000 at 7% p.a. over 25 years

Chapter 1 Credit and borrowing

7

Example

3

E

GC

asio

WORKED

sheet

5 Mr and Mrs French purchase a new lounge suite, which has a cash price of $5500. L Spre XCE ad They purchase the lounge on the following terms: 30% deposit with the balance to be Simple repaid at 9% p.a. flat interest over 2 years. Calculate: interest a the deposit b the balance owing c the interest to be paid am progr –C d the total amount that they pay for the lounge.

–TI

6 Yasmin borrows $5000 from a credit union at a flat interest rate of 8% p.a. to be Interest repaid over 4 years in equal monthly instalments. Calculate: a the interest that Yasmin must pay on the loan b the total amount that Yasmin must repay program GC c the amount of each monthly repayment. 7 Ian borrows $2000 from a pawnbroker at 40% p.a. interest. The loan is to be paid over 1 year in equal weekly payments. a Calculate the interest on the loan. b Calculate the total that Ian must repay. c Calculate Ian’s weekly payment. 8 The Richards family purchase an entertainment system for their home. The total cost of the system is $8000. They buy the system on the following terms: 25% deposit with the balance repaid over 3 years at 12% p.a. flat interest in equal monthly instalments. Calculate: a the deposit b the balance owing c the interest on the loan d the total repayments e the amount of each monthly repayment. 9 Sam buys an electric guitar with a cash price of $1200. He buys the guitar on the following terms: one-third deposit, with the balance at 15% p.a. flat interest over 2 years in equal monthly instalments. Calculate the amount of each monthly repayment. 10 multiple choice The amount of flat rate interest on a loan of $10 000 at 10% p.a. for 2 years is: A $1000 B $2000 C $11 000 D $12 000 11 multiple choice A refrigerator with a cash price of $1800 is bought on the following terms: 20% deposit with the balance paid in 12 equal monthly instalments at 12% p.a. flat interest. The total cost of the refrigerator when purchased on terms is: A $172.80 B $216.00 C $1972.80 D $2016.00

Interest

8

Maths Quest General Mathematics HSC Course

WORKED

Example

4

12 Andy borrows $4000, which is to be repaid over 4 years at $110 per month. Calculate the flat rate of interest that Andy has been charged. 13 Sandra buys a used car with a cash price of $12 000 on the following terms: 20% deposit with the balance paid at $89.23 per week for 3 years. Calculate: a the deposit b the balance owing c the total cost of the car d the flat rate of interest charged. 14 Calculate the flat rate of interest charged on a lounge suite with a cash price of $5000 if it is purchased on the following terms: 15% deposit with the balance paid at $230.21 per month for 2 years.

Computer Application 1 Flat rate interest loan calculator EXCE

et

reads L Sp he

Flat interest

Access the spreadsheet Flat Interest from the Maths Quest General Mathematics HSC Course CD-ROM. This spreadsheet will demonstrate how to calculate a deposit, the total repayments on a loan and the size of each repayment.

Monthly payment calculator Consider a $5000 loan to be repaid at 9% p.a. flat rate interest over 3 years. 1. On the sheet titled ‘Monthly Payments’, in cell B5 enter the amount which has been borrowed ($5000), or the balance owing on a purchase after the deposit has been paid. 2. In cell B7 enter the interest rate as a percentage (9%). 3. In cell B9 enter the number of years over which the loan is to be repaid (3).

Chapter 1 Credit and borrowing

9

4. The total interest paid on the loan will be displayed in cell B11. The formula for this will be displayed in this cell. 5. Cell B13 shows the total amount to be repaid and cell B15 shows the amount of each repayment.

Flat interest rate calculator The worksheet ‘Flat Interest Rate’ will calculate the flat rate of interest charged given the amount of each repayment. Consider a $15 000 loan that is repaid over 5 years at $350 per month. 1. In cell B5 enter the amount borrowed ($15 000). 2. In cell B7 enter the amount of each monthly payment ($350). 3. In cell B9 enter the total number of monthly payments (60).

4. Displayed will be the total amount to be repaid (cell B11), the total interest paid on the loan (cell B13), the amount of interest paid per year (cell B15) and the flat rate of interest (cell B17). Check your answers to the previous exercise using this spreadsheet.

Home loans The biggest loan that most people will ever take out will be for a home. These loans are usually for large amounts of money and are taken over long periods of time. Most commonly they are taken over 10, 15, 20 or 25 years but they can be taken over even longer periods of up to 35 years. Home loans are not charged at a flat rate of interest. The interest on these loans is reducible, which means that the interest is calculated on the amount of money owing on the loan at the time rather than on the amount initially borrowed. This is known as a reducing balance loan. The interest on a home loan is usually calculated at the beginning of each month, and payments are calculated on a monthly basis. So each month interest is added to the loan and a payment is subtracted from the balance owing. The balance increases by the amount of interest and then decreases by the amount of each payment.

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Maths Quest General Mathematics HSC Course

Consider the case of a person who borrows $250 000 to buy a home at 9% p.a. reducible interest. The monthly repayment on this loan is $2500 per month. The interest rate of 9% p.a. converts to 0.75% per month. First month’s interest = 0.75% of $250 000 = $1875 Balance owing = $250 000 + $1875 $2500 = $249 375 In the second month the interest is calculated on the balance owing at the end of the first month. Second month’s interest = 0.75% of $249 375 = $1870.31 Balance owing = $249 375 + $1870.31 = $248 745.31

$2500

The progress of this loan can be followed in the following computer application.

Computer Application 2 Home loan calculator EXCE

et

reads L Sp he

Home loan

Access the spreadsheet Home Loan from the Maths Quest General Mathematics HSC Course CD-ROM. This spreadsheet will allow you to follow the progress of a home loan as it is paid off.

Use the Edit and then the Fill and Down functions on columns A, B, C and D. Look down column D to find when the balance owing becomes 0 or when it becomes negative. At this time the loan will have been fully repaid. Examine other loans by changing the data in C4, C5 and C6.

Chapter 1 Credit and borrowing

11

WORKED Example 5 Mr and Mrs Chan take out a $100 000 home loan at 8% p.a. reducible interest over 25 years. Interest is calculated and added on the first of each month. They make a payment of $775 each month. Calculate: a the interest added after one month b the balance owing after one month. THINK

WRITE

a

a 8% p.a. = 2--- % per month

1 2

Convert 8% p.a. to a monthly rate.

3

Calculate

2 --- % 3

of $100 000 to find the

Interest =

interest for one month. b Add the interest to the principal and subtract the repayment.

2 --- % 3

of $100 000

Interest = $666.67 b Balance owing = $100 000 + $666.67 Balance owing = $99 891.67

$775

the interest in Graphics Calculator tip! Calculate a one-month period We can use the TVM function to calculate the interest for a one-month period but great care needs to be taken. Remembering that the interest is calculated for a number of days, to calculate monthly interest we need to enter n = 365 ÷ 12. Consider the method shown below for worked example 5. 1. From the MENU select TVM.

2. Press F1 to select Simple Interest.

3. n = 365 ÷ 12 (as n is in days) I% = 8 PV = –100000

4. Press F1 (SI) to find the interest for one month.

When interest is calculated every year for such a long period of time, as with many home loans, the amount of money required to pay off such a loan can be a great deal more than the initial loan.

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Maths Quest General Mathematics HSC Course

WORKED Example 6 A loan of $120 000 is paid off at 9% p.a. reducible interest over a period of 25 years. The monthly repayment is $1007.04. Calculate the total amount made in repayments on this loan. THINK 1 2

WRITE

Calculate the number of repayments by multiplying the number of years by 12. Multiply the monthly repayment by the number of repayments.

No. of repayments = 25 ¥ 12 No. of repayments = 300 Total repayments = $1007.04 ¥ 300 Total repayments = $302 112.00

remember 1. The interest on home loans is calculated at a reducible rate. This means that the interest is calculated on the balance owing rather than the initial amount borrowed. 2. Interest is calculated each month; this is then added to the principal and a payment is made. The interest next month is then calculated on the new amount owing. 3. To calculate the total amount to be repaid on a home loan, we multiply the monthly payment by the number of repayments made.

1B SkillS

HEET

1.4

WORKED

Example

Finding values of n and r in financial formulas SkillS

HEET

1.5 Calculating compound interest

EXCE

et

reads L Sp he

Interest

5

Home loans

1 Mr and Mrs Devcich borrow $80 000 to buy a home. The interest rate is 12% p.a. and their monthly payment is $850 per month. a Calculate the interest for the first month of the loan. b Calculate the balance owing at the end of the first month. 2 The repayment on a loan of $180 000 at 7.5% p.a. over a 15-year term is $1668.62 per month. a Calculate the interest for the first month of the loan and the balance owing at the end of the first month. b Calculate the amount by which the balance has reduced in the first month.

Chapter 1 Credit and borrowing

13

c Calculate the interest for the second month of the loan and the balance at the end of the second month. d By how much has the balance of the loan reduced during the second month? 3 The repayment on a loan of $150 000 over a 20-year term at 9.6% p.a. is $1408.01 per month. Copy and complete the table below. Month

Principal ($)

Interest ($)

Balance ($)

1

150 000.00

1200.00

149 791.99

2

149 791.99

3 4 5 6 7 8 9 10 4 Mr and Mrs Roebuck borrow $255 000 to purchase a home. The interest rate is 9% p.a. and over a 25-year term the monthly repayment is $2294.31. a Copy and complete the table below. Month

Principal ($)

Interest ($)

Balance ($)

1

255 000.00

1912.50

254 618.19

2

254 618.19

3 4 5 6 7 8 9 10 11 12

14

Maths Quest General Mathematics HSC Course

b Mr and Mrs Roebuck decide to increase their monthly payment to $2500. Complete the table below. Month

Principal ($)

Interest ($)

Balance ($)

1

255 000.00

1912.50

254 412.50

2

254 412.50

3 4 5 6 7 8 9 10 11 12 c How much less do Mr and Mrs Roebuck owe at the end of one year by increasing their monthly repayment? WORKED

Example

6

5 The repayments on a loan of $105 000 at 8% p.a. reducible interest over 25 years are $810.41 per month. Calculate the total repayments made over the life of the loan. 6 The Taylors borrow $140 000 over 20 years at 9% p.a. a The monthly repayment on this loan is $1259.62. Calculate the total repayments. b The Taylors attempt to pay the loan off quickly by increasing their monthly payment to $1500. The loan is then paid off in 161 months. Calculate the total repayments made under this plan. c How much will the Taylors save by increasing each monthly payment? 7 multiple choice The first month’s interest on a $60 000 home loan at 12% p.a. reducible interest is: A $600 B $7200 C $60 600 D $67 200 8 multiple choice A $95 000 loan at 8% p.a. reducible interest over a 15-year term has a monthly payment of $907.87. The total amount of interest paid on this loan is: A $7600 B $68 416.60 C $114 000 D $163 416.60 9 Mr and Mrs Chakraborty need to borrow $100 000 to purchase a home. The interest rate charged by the bank is 7% p.a. Calculate the total interest paid if the loan is taken over each of the following terms: a $706.78 per month over a 25-year term b $775.30 per month over a 20-year term c $898.83 per month over a 15-year term d $1161.08 per month over a 10-year term.

Chapter 1 Credit and borrowing

15

10 The Smith and Jones families each take out a $200 000 loan at 9.5% p.a. reducible interest. The Smith family repay the loan at $2000 per month and the Jones family repay the loan at $3000 per month. a How much does each family make in repayments in the first year? b Complete the table below for each family. Smith family Month

Principal ($)

Interest ($)

Balance ($)

1

200 000.00

1583.33

199 583.33

2

199 583.33

3 4 5 6 7 8 9 10 11 12 Jones family Month

Principal ($)

Interest ($)

Balance ($)

1

200 000.00

1583.33

198 583.33

2

198 583.33

3 4 5 6 7 8 9 10 11 12 c After one year how much less does the Jones family owe than the Smith family?

16

Maths Quest General Mathematics HSC Course

1 1 Calculate the amount of flat rate interest payable on a loan of $1500 at 14% p.a. to be repaid over 2 years. 2 Calculate the amount of flat rate interest payable on a loan of $2365 at 19.2% p.a. to be repaid over 2 1--- years. 2

3 Calculate the total repayments on a loan of $5000 at 13.5% p.a. to be repaid over 3 years. 4 Susan buys a lounge suite on terms. The cash price of the lounge is $6500 and she pays a 15% deposit. Calculate the amount of the deposit. 5 Calculate the balance that Susan owes on the lounge suite. 6 Calculate the interest that Susan will pay at 17% p.a. flat rate interest for a period of 3 years. 7 Calculate the total amount that Susan will have to repay. 8 Calculate the monthly repayment that Susan will need to make. 9 Harry and Sally borrow $164 000 to purchase a home. The interest rate is 12% p.a. Calculate the amount of interest payable for the first month. 10 A $175 000 loan that is repaid over 25 years has a monthly repayment of $1468.59. Calculate the total amount of interest that is paid on this loan.

The cost of a loan Because of the different ways that interest can be calculated, the actual interest rate quoted may not be an accurate guide to the cost of the loan. By using a flat rate of interest, a lender can quote an interest rate less than the equivalent reducible interest rate. To compare flat and reducible rates of interest, we need to calculate the effective rate of interest for a flat rate loan. The effective rate of interest is the equivalent rate of reducible interest for a flat rate loan. The formula for effective rate of interest is: ( 1 + r )n – 1 E = ---------------------------n where E = effective rate of interest, expressed as a decimal r = stated rate of flat interest expressed as a decimal n = term of the loan in years Note: This formula for effective rate of interest is not on your formula sheet. This does not mean that you have to memorise it as the formula will be given to you as a part of any question that requires you to use it.

Chapter 1 Credit and borrowing

17

WORKED Example 7 Andrea takes out an $8000 loan for a car over 5 years at 6% p.a. flat rate interest. Calculate the effective rate of interest charged on the loan. THINK

WRITE

1

Write the formula.

2

Substitute r = 0.06 and n = 5.

3

Calculate.

4

Write the interest rate as a percentage.

( 1 + r )n – 1 E = ---------------------------n ( 1.06 ) 5 – 1 E = -------------------------5 E = 0.068 The effective rate of interest is 6.8% p.a.

A loan with a reducible rate of interest can be compared to a flat rate of interest if we are able to calculate the total repayments made over the term of the loan.

WORKED Example 8 An $85 000 loan at 10% p.a. reducible interest is to be repaid over 15 years at $913.41 per month. a Calculate the total repayments on the loan. b Calculate the total amount of interest paid. c Calculate the equivalent flat rate of interest on this loan. THINK

WRITE

a Multiply the monthly repayments by the number of months taken to repay the loan.

a Total repayments = $913.41 180 Total repayments = $164 413.80

b Subtract the initial amount borrowed from the total repayments.

b Interest = $164 413.80 Interest = $79 413.80

c

c Annual interest = $79 413.80 ÷ 15 Annual interest = $5294.25

1

Calculate the amount of interest paid per year.

2

Write the yearly interest as a percentage of the amount borrowed.

$85 000

$5294.25 Flat interest rate = ---------------------$85 000 Flat interest rate = 6.2% p.a.

100%

The most accurate way to compare loans is to calculate the total repayments made in each loan.

18

Maths Quest General Mathematics HSC Course

WORKED Example 9 Allison borrows $6000 and has narrowed her choice of loans down to two options. Loan A: At 8% p.a. flat rate interest over 4 years to be repaid at $165.00 per month. Loan B: At 12% p.a. reducible interest over 3 years to be paid at $199.29 per month. Which of the two loans would cost Allison less? THINK 1 2 3

WRITE

Calculate the total repayments on Loan A. Calculate the total repayments on Loan B. Write a conclusion.

Loan A repayments = $165.00 48 Loan A repayments = $7920 Loan B repayments = $199.29 36 Loan B repayments = $7174.44 Loan B would cost $745.56 less than Loan A.

In the above example Allison should take Loan B even though it has a much higher advertised interest rate. This of course would depend upon Allison’s ability to meet the higher monthly payments. Generally the more quickly that you can pay off a loan the cheaper the loan will be. The savings are particularly evident when examining home loans. Some home loans that offer a lower interest rate allow for you to make only the minimum monthly repayment. This will maximise the amount of interest that the customer will pay. If a person can afford to pay more than the minimum amount, they may be better off over time by paying a slightly higher rate of interest and paying the loan off over a shorter period of time.

WORKED Example 10 Mr and Mrs Beasley need to borrow $100 000 and have the choice of two home loans. Loan X: 6% p.a. over 25 years with a fixed monthly repayment of $644.30. No extra repayments are allowed on this loan. Loan Y: 7% p.a. over 25 years with a minimum monthly payment of $706.78. Mr and Mrs Beasley believe they can afford to pay $800 per month on this loan. If they do, the loan will be repaid in 18 years and 9 months. Which loan would you recommend? THINK 1 2 3

WRITE

Calculate the total repayments on Loan X. Calculate the total repayments on Loan Y. Make a recommendation.

Loan X repayments = $644.30 300 = $193 290 Loan Y repayments = $800 225 = $180 000 Mr and Mrs Beasley should choose Loan Y as they will save $13 290 provided they can continue to pay $800 per month.

With loans such as the one in the above example, the savings depend upon the ability to make the extra repayments. If this is doubtful, Loan X would have been the safer option.

Chapter 1 Credit and borrowing

19

The other factor to consider when calculating the cost of a loan is fees. Many loans have a monthly management fee attached to them. This will need to be calculated into the total cost and may mean that a loan with a slightly higher interest rate but no fee may be a cheaper option.

remember 1. The actual cost of a loan is calculated by the total cost in repaying the loan. The interest rate is a guide but not the only factor in calculating cost. 2. A loan that is quoted at a flat rate of interest can be compared to a reducible rate of interest only by calculating the effective rate of interest on the flat rate loan. The effective rate of interest is the equivalent reducible rate of interest and is found using the formula: ( 1 + r )n – 1 E = ---------------------------n 3. By calculating the total repayments on a loan, we can calculate the equivalent flat rate of interest paid on the loan. 4. A loan that is repaid over a shorter period of time will usually cost less than one where the repayments are made over the full term of the loan. 5. The flexibility of a loan, which includes factors such as whether extra repayments can be made, is important when considering the cost of a loan. 6. When calculating the cost of a loan, any ongoing fees need to be calculated.

WORKED

Example

1 A $15 000 loan is to be repaid at 8% p.a. flat rate interest over a 10-year term. ( 1 + r )n – 1 Use the formula E = ---------------------------- to calculate the effective rate of interest. n

Substitution into a formula

3 A bank offers loans at 8% p.a. flat rate of interest. Calculate the effective rate of interest for a loan taken over: a 2 years b 3 years c 4 years d 5 years e 10 years f 20 years. WORKED

Example

8

SkillS

4 An $85 000 home loan at 9% p.a reducible interest is to be repaid over 25 years at $713.32 per month. a Calculate the total repayments on the loan. b Calculate the total amount of interest paid. c Calculate the equivalent flat rate of interest on the loan. 5 Calculate the equivalent flat rate of interest paid on a $115 000 loan at 12% p.a. reducible interest to be repaid over 30 years at $1182.90 per month.

L Spre XCE ad

Effective rate of interest

sheet

2 Calculate the effective rate of interest on each of the following flat rate loans. a 10% p.a. over 4 years b 8% p.a. over 2 years c 12% p.a. over 5 years d 7.5% p.a. over 10 years e 9.6% p.a. over 6 years

1.6

HEET

7

The cost of a loan

E

1C

20

Maths Quest General Mathematics HSC Course

WORKED

Example

9

6 Kim borrows $12 000 for a holiday to South-East Asia. She is faced with a choice of two loans. Loan I: At 10% p.a. flat rate of interest over 2 years to be repaid at $600 per month. Loan II: At 12.5% p.a. reducible interest over 3 years to be repaid at $401.44 per month. Which loan will cost Kim the least money? 7 Calculate the total cost of repaying a loan $100 000 at 8% p.a. reducible interest: a over 25 years with a monthly repayment $771.82 b over 20 years with a monthly repayment $836.44 c over 10 years with a monthly repayment $1213.28.

WORKED

Example

10

of of of of

8 Masako and Toshika borrow $125 000 for their home. They have the choice of two loans. Loan 1: A low interest loan at 7% p.a. interest over 25 years with fixed repayments of $833.47 per month. Loan 2: A loan at 7.5% p.a interest over 25 years with minimum repayments of $923.74 per month. Masako and Toshika believe they can afford to pay $1000 per month. If they do, Loan 2 will be repaid in 20 years and 4 months. Which loan should they choose if they could afford to pay the extra each month? 9 multiple choice A loan can be taken out at 8% p.a. flat interest or 9% p.a. reducible interest. Using the ( 1 + r )n – 1 formula E = ---------------------------- , the number of years of the loan (n) after which the effective n rate of interest on the flat rate loan becomes greater than the reducible rate loan is: A 2 years B 3 years C 4 years D 5 years 10 Glenn and Inge are applying for a $150 000 loan to be repaid over 25 years. a Bank A charges 7.8% p.a. interest, no fees, with the loan to be repaid at $1137.92 per month. Calculate the total cost of this loan. b Bank B charges 7.6% p.a. interest, a $600 loan application fee, a $5 per month management fee and repayments of $1118.26 per month. Calculate the total cost of this loan. 11 multiple choice A $50 000 loan is to be taken out. Which of the following loans will have the lowest total cost? A 5% p.a. flat rate interest to be repaid over 10 years B 8% p.a. reducible interest to be repaid over 10 years at $606.64 per month C 6% p.a. reducible interest to be repaid over 12 years at $487.93 per month D 6.5% p.a. reducible interest to be repaid over 10 years at $567.74 per month, with a $600 loan application fee and $8 per month account management fee

Chapter 1 Credit and borrowing

21

12 A home loan of $250 000 is taken out over a 20-year term. The interest rate is 9.5% p.a. and the monthly repayments are $2330.33.

b Calculate the equivalent flat rate of interest on the loan. (Consider the extra payments as part of the interest.) c If the loan is repaid at $3000 per month, it will take 11 1--- years to repay the loan. 2 Calculate the equivalent flat rate of interest if this repayment plan is followed.

Graphics Calculator tip! Loan repayment function Your Casio graphics calculator can calculate the amount of each monthly repayment on a home loan when given the term of the loan and the interest rate. The PMT function, which is under the compound interest menu, allows for such calculations to be made. Consider a loan of $250 000 to be repaid over 25 years at 8% p.a. with interest added and repayments made monthly. We wish to find the amount of each monthly repayment. 1. From the MENU select TVM.

2. Press F2 to select Compound Interest.

3. Enter the following settings. n = 25 12 I% = 8 PV = 250000 PMT = 1 FV = 0 P/Y = 12 (payments per year) C/Y = 12 (You will need to scroll to see this.) 4. Press F4 (PMT) to find the amount of each monthly repayment, which will be displayed as a negative.

Work

a The mortgage application fee on this loan was $600 and there is a $10 per month account management fee. Calculate the total cost of repaying this loan. T SHEE

1.1

22

Maths Quest General Mathematics HSC Course

Researching home loans 1 Suppose that you wish to borrow $100 000 to buy a home. Go to a bank or other lender and gather the following information. a The annual interest rate b The loan application fee and any other costs such as stamp duty, legal costs etc. associated with establishing the loan c Is there a monthly account keeping or management fee? d The monthly repayment if the loan is repaid over: i 15 years ii 20 years iii 25 years e The total cost of repaying the loan in each of the above examples 2 There are many ways that people can reduce the overall cost of repaying a mortgage. Research and explain why people are able to save money by adopting the following repayment strategies. a Repaying the loan fortnightly or weekly instead of monthly b Using an account where the whole of a person’s net pay is deposited on the mortgage and then a redraw is used to meet living expenses

Credit cards Credit cards are the most common line of day-to-day credit that most people use. A credit card works as a pre-approved loan up to an amount agreed upon by the customer and the bank. The card can then be used until the amount of the debt reaches this limit. As with other types of loan, the bank charges interest upon the amount that is owed on the card and repayments must be made monthly. The way in which the interest is calculated varies with different types of credit cards. Some cards have interest charged from the day on which the purchase was made. Others have what is called an interest-free period. This means that a purchase that is made will appear on the next monthly statement. Provided that this amount is paid by the due date, no interest is charged. Hence, the customer can repay the loan within a maximum of 55 days and be charged no interest. Generally, credit cards without an interest-free period have a lower interest rate than those with an interest-free period. These cards, however, generally attract an annual fee. This annual fee can in some cases be waived if a certain amount is spent on the card over the year. The minimum monthly repayment on most credit cards is 5% of the outstanding balance, or $10, whichever is greater.

WORKED Example 11

On Trevor’s credit card statement he has an outstanding balance of $1148.50. The minimum monthly payment is 5% of the outstanding balance, or $10, whichever is greater. Calculate the minimum repayment that Trevor must make. THINK 1 Calculate 5% of the outstanding balance. 2 Decide which repayment is greater and give a written answer.

WRITE 5% of $1148.50 = $57.43 The minimum repayment is $57.43.

Chapter 1 Credit and borrowing

23

Credit card interest is quoted as an annual amount but is added monthly. To calculate the interest due, calculate one month’s interest on the outrstanding balance..

WORKED Example 12 The outstanding balance on a credit card is $2563.75. If the full balance is not paid by the due date, one month’s interest will be added at a rate of 18% p.a. Calculate the amount of interest that will be added to the credit card. THINK

WRITE

Use the simple interest formula to calculate one month’s interest.

I = Prn I = $2563.75 I = $38.46

0.18

1 -----12

In practice, most credit cards calculate interest on the outstanding balance at a daily rate and then add the interest monthly. If a credit card advertises its interest rate as 18% p.a., the daily rate is 0.049 315%. To work out the interest, you will need to count the number of days that the credit card has each different balance over the month.

WORKED Example 13 An extract from a credit card statement is shown below. Interest rate = 15% p.a. Daily rate = 0.041 096% Date

Credit

Debit

Balance

1 June 10 June

$900 $400 – repayment

$500

15 June

$350 – purchase

$850

22 June

$140 – purchase

$990

1 July

??? – interest

Calculate the interest that will be due for the month of June. THINK 1 2 3 4 5

For 1 June – 9 June inclusive (9 days), the balance owing is $900. Calculate the interest. For 10 June – 14 June inclusive (5 days), the balance owing is $500. Calculate the interest. For 15 June – 21 June inclusive (7 days), the balance owing is $850. Calculate the interest. For 22 June – 30 June inclusive (9 days), the balance owing is $990. Calculate the interest. Add each amount of interest to calculate the total interest for the month.

WRITE I = 0.041 096% of $900 9 I = $3.33 I = 0.041 096% of $500 5 I = $1.03 I = 0.041 096% of $850 7 I = $2.45 I = 0.041 096% of $990 9 I = $3.66 Total interest = $3.33 + $1.03 Total interest = + $2.45 + $3.66 Total interest = $10.47

24

Maths Quest General Mathematics HSC Course

Graphics Calculator tip! Calculating interest on a daily basis When doing this type of question where we need to consider interest calculated on a daily basis the TVM mode of your calculator is very useful. Consider the method shown below for worked example 13. 1. From the MENU of your calculator select TVM.

2. Press F1 to select Simple Interest.

3. For 9 days the balance is $900, so enter: n=9 I% = 15 PV = –900

4. Press F1 (SI) to get the interest for these 9 days.

Interest = $3.33 5. For 5 days the balance is $500. Press EXIT to return to the previous screen; change the values of n and PV. n=5 I% = 15 PV = –500

Interest = $1.03

Then press F1 for the simple interest. 6. For 7 days the balance is $850. Press EXIT to return to the previous screen; change the values of n and PV. n=7 I% = 15 PV = –850

Then again press F1 for the simple interest.

Interest = $2.45

Chapter 1 Credit and borrowing

7. For 9 days the balance is $990. Press EXIT to return to the previous screen; change the values of n and PV. n=9 I% = 15 PV = –990 Then again press F1 for the simple interest. 8. Add each amount of interest to find the total amount of interest for the month.

25

Interest = $3.66

Total interest = $3.33 + $1.03 + $2.45 + $3.66 = $10.47

When deciding which credit card is most suitable for your needs, consider if you will generally be able to pay most items off before the interest-free period expires. The total cost in interest over a year will vary according to the repayment pattern.

WORKED Example 14

Kerry has a credit card with an interest-free period and interest is then charged on the outstanding balance at a rate of 18% p.a. Kerry pays a $1200 bill for her council rates on her credit card. a Kerry pays $600 by the due date. What is the outstanding balance on the card? b Calculate the interest Kerry must then pay for the second month. c An alternative credit card charges 12% p.a. interest with no interest-free period. Calculate the interest that Kerry would have been charged on the first month. d Calculate the balance owing after Kerry pays $600 then calculate the interest for the second month. e Which credit card would be the cheapest to use for this bill? THINK

WRITE

a Subtract the repayment from the balance.

a Balance owing = $1200 Balance owing = $600

b Use the simple interest formula to calculate one month’s interest.

b I = Prn = $600 0.18 = $9.00

c Use the simple interest formula to calculate the first month’s interest.

c I = Prn = $1200 0.12 = $12.00

d

d Balance owing = $1200 + $12 = $612 I = Prn 1 = $612 0.12 ----12 = $6.12

1 2

Add the interest to the amount of the bill and subtract the repayment. Use the simple interest formula to calculate the second month’s interest.

e Add the two months of interest together for the second card and compare with the interest for the first card.

$600

1 -----12

1 -----12

$600

e The interest on the second card is $18.12 and therefore the card with the interest-free period is cheaper in this case.

26

Maths Quest General Mathematics HSC Course

remember 1. A credit card is a source of an instant loan to the card holder. 2. The card is repaid monthly with the minimum payment usually 5% of the outstanding balance, or $10, whichever is the greater. 3. There are many different types of credit card. The main difference between them is that some have an interest-free period while others charge interest from the date of purchase. 4. Cards without an interest-free period generally have a lower rate of interest than those with an interest-free period. 5. The interest on a credit card is usually calculated as a daily rate. This is found by dividing the annual rate by 365. 6. The TVM function on the graphics calculator can be used to calculate the monthly interest on a credit card. 7. To calculate the cheaper credit card, we need to consider the repayment plan that would be used.

1D

Credit cards

1 Roy has a credit card with an outstanding balance of $2730. Calculate the minimum payment if he must pay 5% of the balance, or $10, whichever is greater. 11 2 The minimum monthly repayment on a credit card is 5% of the balance, or $10, whichever is greater. Calculate the minimum monthly repayment on a balance of: a $3500 b $1194.50 c $492.76 d $150 e $920.52.

WORKED

Example

3 Leonie has a credit card with an outstanding balance of $1850. If the interest rate is 18% p.a., calculate the amount of interest that Leonie will be charged for one month if 12 the balance is not paid by the due date.

WORKED

Example

4 Hassim buys a refrigerator for $1450 with his credit card. The card has no interest-free period and interest is charged at a rate of 15% p.a. Calculate one month’s interest on this purchase. 5 Michelle has a $2000 outstanding balance on her credit card. The interest rate charged is 21% p.a. on the balance unpaid by the due date. a If Michelle pays $200 by the due date, calculate the balance owing. b Calculate the interest that Michelle will owe for the next month. c What will be the balance owing on Michelle’s next credit card statement? d What will be the total amount owing on the credit card after another month’s interest is added? 6 Chandra has a credit card which charges interest at a rate of 12% p.a. but has no interest-free period. He makes a purchase of $1750 on the credit card. a After one month Chandra’s credit card statement arrives. What will be the outstanding balance on the statement? b The minimum repayment will be 5% of the outstanding balance. Calculate the amount that Chandra will owe if he makes only the minimum payment. c In the next month Chandra makes purchases totalling $347.30. Calculate the interest charged and the balance owing for the next month’s statement.

Chapter 1 Credit and borrowing

WORKED

Example

13

27

7 An extract of a credit card statement is shown below. Take 1 year = 365.25 days. Interest rate = 18% p.a. Date

Daily rate = 0.049 28%

Credit ($)

Debit ($)

1 July

256.40

10 July 20 July

Balance ($)

40 – purchase 40 – repayment

1 August

??? – interest

a Complete the balance column. Calculate the balance owing on 10 July and 20 July. b Calculate the interest due on 1 August and the balance on that date. 8 Study the credit card statement below. Interest rate = 16.5% p.a. Date

Daily rate = ______

Credit ($)

Debit ($)

1 Jan. 8 Jan.

1548.50 500 – repayment

15 Jan.

399 – purchase

1 Feb. 8 Feb. 1 March

Balance ($)

??? – interest ??? – repayment ??? – interest

a Calculate the daily rate of interest, correct to 4 decimal places (take 1 year = 365.25 days). b Calculate the interest added to the account on 1 February. c On 8 February the minimum repayment of 5% is made. Calculate the amount of this repayment. d Calculate the outstanding balance on the account on 1 March. 9 Kai has two credit cards. One has an interest-free period and interest is then charged on the outstanding balance at a rate of 18% p.a. The other has no interest-free period with 14 interest added from the date of purchase at a rate of 14% p.a. Kai has $1500 worth of bills to pay in the coming month and intends to use one of the cards to pay them, then pay the balance off in monthly instalments of $500. a If Kai uses the card with the interest-free period and pays $500 by the due date, what is the outstanding balance on the card? b Calculate the interest Kai must then pay for the second month. c Calculate the balance owing at the end of the second month and the balance owing at the end of the third month, at which time Kai pays off the entire balance. d Calculate the interest payable in the first month if Kai uses the card without the interest-free period. e Calculate the balance owing after Kai pays $500 then calculate the interest for the second month. f Calculate the balance owing at the end of the second month and the balance owing at the end of the third month, at which time Kai pays off the entire balance. g Which card should Kai use for these bills?

WORKED

Example

28

Maths Quest General Mathematics HSC Course

Researching credit cards Find out about the costs associated with two credit cards. One of the cards should have an interest-free period and the other no interest-free period. Find out: 1 if there is an annual fee associated with having the card 2 the interest rate on each card 3 the minimum monthly payment to be made on each card 4 what credit limits apply to a first-time credit card holder 5 what benefits such as ‘Fly-Buys’ or ‘Frequent Flyer Points’ can be obtained from use of the card 6 any other relevant information about the card.

2 1 Calculate the amount of flat rate interest payable on a loan of $4500 at 21% p.a. over a 3 year term. 2 A loan of $2000 is repaid over 1 year at a rate of $100 per week. Calculate the rate of interest charged on the loan. 3 A loan of $120 000 at 11% p.a. reducible over 20 years is repaid at $1238.63 per month. The bank also charges an $8 per month account management fee. Calculate the total cost of repaying the loan. 4 A loan of $5000 is advertised at a rate of 9% p.a. flat rate interest for a term of ( 1 + r )n – 1 4 years. Use the formula E = ---------------------------- to calculate the effective rate of interest on n this loan (correct to 1 decimal place). 5 A loan of $10 000 at 11% p.a. reducible interest is repaid over 4 years at a rate of $258.46 per month. Calculate the equivalent flat rate of interest charged on the loan (correct to 1 decimal place). 6 With reference to credit cards, what is meant by the term interest-free period? 7 The minimum repayment on a credit card is 5% or $10, whichever is greater. Calculate the minimum repayment for July that is to be made on a card with an outstanding balance of $3297.50. 8 On the credit card in question 7, a repayment of $500 is made by the due date. Calculate the interest that will be charged for August at the rate of 18% p.a. 9 An alternative credit card with no interest-free period has an interest rate of 12% p.a. Calculate the interest on the above credit card for July at this rate. 10 Calculate the total interest that would have been charged for 2 months assuming a $500 payment was made on 1 August.

Chapter 1 Credit and borrowing

29

Loan repayments With a reducing balance loan, an amount of interest is added to the principal each month and then a repayment is made which is then subtracted from the outstanding balance. Consider the case below of a $2000 loan at 15% p.a. to be repaid over 1 year in equal monthly instalments of $180.52. Month

Opening balance

Interest

Closing balance

1

$2000.00

$25.00

$1844.48

2

$1844.48

$23.06

$1687.02

3

$1687.02

$21.09

$1527.59

4

$1527.59

$19.09

$1366.17

5

$1366.17

$17.08

$1202.73

6

$1202.73

$15.03

$1037.25

7

$1037.25

$12.97

$ 869.70

8

$ 869.70

$10.87

$ 700.05

9

$ 700.05

$ 8.75

$ 528.29

10

$ 528.29

$ 6.60

$ 354.37

11

$ 354.37

$ 4.43

$ 178.29

12

$ 178.29

$ 2.23

−$

0.00

The actual calculation of the amount to be repaid each month to pay off the loan plus interest in the given period of time is beyond this course. The most practical way to find the amount of each monthly repayment is to use a table of repayments.

30

Maths Quest General Mathematics HSC Course

Monthly repayment per $1000 borrowed Interest rate Year

5%

6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

1

$85.61 $86.07 $86.53 $86.99 $87.45 $87.92 $88.38 $88.85 $89.32 $89.79 $90.26

2

$43.87 $44.32 $44.77 $45.23 $45.68 $46.14 $46.61 $47.07 $47.54 $48.01 $48.49

3

$29.97 $30.42 $30.88 $31.34 $31.80 $32.27 $32.74 $33.21 $33.69 $34.18 $34.67

4

$23.03 $23.49 $23.95 $24.41 $24.89 $25.36 $25.85 $26.33 $26.83 $27.33 $27.83

5

$18.87 $19.33 $19.80 $20.28 $20.76 $21.25 $21.74 $22.24 $22.75 $23.27 $23.79

6

$16.10 $16.57 $17.05 $17.53 $18.03 $18.53 $19.03 $19.55 $20.07 $20.61 $21.15

7

$14.13 $14.61 $15.09 $15.59 $16.09 $16.60 $17.12 $17.65 $18.19 $18.74 $19.30

8

$12.66 $13.14 $13.63 $14.14 $14.65 $15.17 $15.71 $16.25 $16.81 $17.37 $17.95

9

$11.52 $12.01 $12.51 $13.02 $13.54 $14.08 $14.63 $15.18 $15.75 $16.33 $16.92

10

$10.61 $11.10 $11.61 $12.13 $12.67 $13.22 $13.78 $14.35 $14.93 $15.53 $16.13

11

$ 9.86 $10.37 $10.88 $11.42 $11.96 $12.52 $13.09 $13.68 $14.28 $14.89 $15.51

12

$ 9.25 $ 9.76 $10.28 $10.82 $11.38 $11.95 $12.54 $13.13 $13.75 $14.37 $15.01

13

$ 8.73 $ 9.25 $ 9.78 $10.33 $10.90 $11.48 $12.08 $12.69 $13.31 $13.95 $14.60

14

$ 8.29 $ 8.81 $ 9.35 $ 9.91 $10.49 $11.08 $11.69 $12.31 $12.95 $13.60 $14.27

15

$ 7.91 $ 8.44 $ 8.99 $ 9.56 $10.14 $10.75 $11.37 $12.00 $12.65 $13.32 $14.00

16

$ 7.58 $ 8.11 $ 8.67 $ 9.25 $ 9.85 $10.46 $11.09 $11.74 $12.40 $13.08 $13.77

17

$ 7.29 $ 7.83 $ 8.40 $ 8.98 $ 9.59 $10.21 $10.85 $11.51 $12.19 $12.87 $13.58

18

$ 7.03 $ 7.58 $ 8.16 $ 8.75 $ 9.36 $10.00 $10.65 $11.32 $12.00 $12.70 $13.42

19

$ 6.80 $ 7.36 $ 7.94 $ 8.55 $ 9.17 $ 9.81 $10.47 $11.15 $11.85 $12.56 $13.28

20

$ 6.60 $ 7.16 $ 7.75 $ 8.36 $ 9.00 $ 9.65 $10.32 $11.01 $11.72 $12.44 $13.17

21

$ 6.42 $ 6.99 $ 7.58 $ 8.20 $ 8.85 $ 9.51 $10.19 $10.89 $11.60 $12.33 $13.07

22

$ 6.25 $ 6.83 $ 7.43 $ 8.06 $ 8.71 $ 9.38 $10.07 $10.78 $11.50 $12.24 $12.99

23

$ 6.10 $ 6.69 $ 7.30 $ 7.93 $ 8.59 $ 9.27 $ 9.97 $10.69 $11.42 $12.16 $12.92

24

$ 5.97 $ 6.56 $ 7.18 $ 7.82 $ 8.49 $ 9.17 $ 9.88 $10.60 $11.34 $12.10 $12.86

25

$ 5.85 $ 6.44 $ 7.07 $ 7.72 $ 8.39 $ 9.09 $ 9.80 $10.53 $11.28 $12.04 $12.81 The table shows the monthly repayment on a $1000 loan at various interest rates over various terms. To calculate the repayment on a loan, we simply multiply the repayment on $1000 by the number of thousands of dollars of the loan.

Chapter 1 Credit and borrowing

31

WORKED Example 15 Calculate the monthly repayment on a loan of $85 000 at 11% p.a. over a 25-year term. THINK 1 2

WRITE

Look up the table to find the monthly repayment on $1000 at 11% p.a. for 25 years. Multiply this amount by 85.

Monthly repayment = $9.80 Monthly repayment = $833

85

This table can also be used to make calculations such as the effect that interest rate rises will have on a home loan.

WORKED Example 16 The Radley family borrow $160 000 for a home at 8% p.a. over a 20-year term. They repay the loan at $1400 per month. If the interest rate rises to 9%, will they need to increase their repayment and, if so, by how much? THINK 1 2 3

WRITE

Look up the table to find the monthly repayment on $1000 at 8% p.a. for 20 years. Multiply this amount by 160. If this amount is greater than $1400, state the amount that the repayment needs to rise.

Monthly repayment = $9.00 160 Monthly repayment = $1440.00 The Radley family will need to increase their monthly repayments by $40.

remember 1. The amount of each monthly repayment is best determined by using a table of repayments. 2. The amount of each repayment is calculated by multiplying the monthly repayment on a $1000 loan by the number of thousands of the loan.

1E

Loan repayments

1 Use the table of repayments on page 30 to calculate the monthly repayment on a 1.7 S killS $75 000 loan at 7% p.a. over a 15-year term. 15 2 Use the table of repayments to calculate the monthly repayment on each of the Reading tables following loans. a $2000 at 8% p.a. over a 2-year term L Spre b $15 000 at 13% p.a. over a 5-year term XCE ad c $64 000 at 15% p.a. over a 25-year term Reducing d $100 000 at 12% p.a. over a 20-year term balance e $174 000 at 9% p.a. over a 22-year term loans

WORKED

Example

sheet

E

HEET

32

Maths Quest General Mathematics HSC Course

3 Jenny buys a computer for $4000 on the following terms: 10% deposit with the balance paid in equal monthly instalments over 3 years at an interest rate of 14% p.a. a Calculate Jenny’s deposit. b Calculate the balance owing on the computer. c Use the table of repayments to calculate the amount of each monthly repayment. 4 Mr and Mrs Dubois borrow $125 000 over 20 years at 10% p.a. to purchase a house. They repay the loan at a rate of $1500 per month. If the interest rate rises to 12% p.a., 16 will Mr and Mrs Dubois need to increase the size of their repayments and, if so, by how much?

WORKED

Example

5 Mr and Mrs Munro take out a $180 000 home loan at 9% p.a. over a 25-year term. a Calculate the amount of each monthly repayment. b After 5 years the balance on the loan has been reduced to $167 890. The interest rate then rises to 10% p.a. Calculate the new monthly repayment required to complete the loan within the existing term. 6 A bank will lend customers money only if they believe the customer can afford the repayments. To determine this, the bank has a rule that the maximum monthly repayment a customer can afford is 25% of his or her gross monthly pay. Darren applies to the bank for a loan of $62 000 at 12% p.a. over 15 years. Darren has a gross annual salary of $36 000. Will Darren’s loan be approved? Use calculations to justify your answer. 7 Tracey and Barry have a combined gross income of $84 000. a Calculate Tracey and Barry’s gross monthly income. b Using the rule applied in the previous question, what is the maximum monthly repayment on a loan that they can afford? c If interest rates are 11% p.a., calculate the maximum amount (in thousands) that they could borrow over a 25-year term. 8 Mr and Mrs Yousef borrow $95 000 over 25 years at 8% p.a. interest. a Calculate the amount of each monthly repayment on the loan. b Mr and Mrs Yousef hope to pay the loan off in a much shorter period of time. By how much will they need to increase the monthly repayment to pay the loan off in 15 years? 9 Mr and Mrs Bath borrow $375 000 at 8% p.a. reducible over a 25-year term, with repayments to be made monthly. a Calculate the amount of each monthly repayment. b Calculate the total amount that Mr and Mrs Bath will repay over the term of the loan. c What is the total amount of interest that Mr and Mrs Bath will pay on the loan? d Calculate the average amount of interest that Mr and Mrs Bath will pay each year. e Calculate the equivalent flat rate of interest by expressing your answer to part d as a percentage of the amount borrowed.

33

10 A loan of $240 000 is taken out over a 25-year term at an interest rate of 7% p.a. reducible. a Calculate the amount of each monthly repayment. b Calculate the total repayments made on the loan. c Calculate the amount of interest paid on the loan. d Find the equivalent flat rate of interest. e By following steps a to d above calculate the equivalent flat rate of interest if the term of the loan is: ii 20 years ii 15 years.

Work

Chapter 1 Credit and borrowing

T SHEE

1.2

Computer Application 3 Loan repayments

1. $80 000 2. $50 000

E

L Spre XCE ad

Loan repayments

3. $20 000. Next, change the amount borrowed in the spreadsheet to $200 000. Does it take the same length of time for the outstanding balance to be halved? Change the interest rate to 12% p.a. and the amount borrowed back to $100 000. Does it still take the same length of time for the balance to be halved? Experiment with different loans and look for a pattern in the way in which the balance of the loan reduces.

sheet

Access the spreadsheet Loan Repayments from the Maths Quest General Mathematics HSC Course CD-ROM. This spreadsheet shows the graph of a home loan of $100 000 at 9% p.a. that is repaid over 25 years. Use the graph to determine how long it takes for the outstanding balance to reduce to:

34

Maths Quest General Mathematics HSC Course

summary Flat rate interest • A flat rate loan is one where interest is calculated based on the amount initially borrowed. • Flat rate loans have the interest calculated using the simple interest formula: I = Prn • The total repayments on a flat rate loan are calculated by adding the interest to the amount borrowed. • The monthly or weekly repayments on a flat rate loan are calculated by dividing the total repayments by the number of weeks or months in the term of the loan.

Home loans • The interest of home loans is calculated at a reducible rate. This means that the interest is calculated on the outstanding balance at the time and not on the initial amount borrowed. • The interest on home loans is usually calculated and added monthly while repayments are calculated on a monthly basis. • To calculate the total cost of a home loan, we multiply the amount of each monthly payment by the number of payments.

The cost of a loan • To compare a flat rate loan with a reducing balance loan, the equivalent reducing balance interest rate can be calculated using the formula: ( 1 + r )n – 1 E = ---------------------------n • When comparing two or more loans, the most accurate comparison is done by calculating the total cost of repaying the loan. • A loan that is repaid over a shorter period of time will generally cost less even if the interest rate may be slightly higher. • The flexibility of loan repayments is an important consideration when calculating the cost of a loan. • When calculating the cost of a loan, fees such as application fee and account management fees must be considered along with the interest payable.

Credit cards • A credit card is a pre-approved loan up to a certain amount called the credit limit. • There are many kinds of credit cards and the most important difference is that some cards have an interest-free period while others attract interest from the date of purchase. • Credit cards without an interest-free period generally have a lower rate of interest than those with an interest-free period.

Chapter 1 Credit and borrowing

35

• Each credit card will have a monthly statement and will require a minimum payment each month. • When evaluating the best credit card for your circumstances, you need to consider if you will be able to pay most bills by the due date and consider any fees attached to the card.

Loan repayments • The amount of each monthly repayment is best calculated using a table of monthly repayments. • The monthly repayment on a $1000 loan at the given rate over the given term is then multiplied by the number of thousands of the loan to find the size of each repayment.

36

Maths Quest General Mathematics HSC Course

CHAPTER review 1A

1 Calculate the amount of flat rate interest that will be paid on each of the following loans. a $8000 at 7% p.a. for 2 years b $12 500 at 11.5% p.a. for 5 years c $2400 at 17.8% p.a. for 3 years d $800 at 9.9% p.a. over 6 months e $23 400 at 8.75% p.a. over 6 years

1A

2 Calculate the total repayments made on a loan of $4000 at 23% p.a. flat rate interest to be repaid over 3 years.

1A

3 Noel borrows $5600 at 7.6% p.a. flat rate interest to be repaid in monthly instalments over 3 years. Calculate the amount of each monthly instalment.

1A

4 Shane borrows $9500 to purchase a new car. He repays the loan over 4 years at a rate of $246.60 per month. Calculate the flat rate of interest charged on the loan.

1B

5 Mr and Mrs Smith borrow $125 000 to purchase a home. The interest rate is 12% p.a. and the monthly repayments are $1376.36. Calculate: a the first month’s interest on the loan b the balance of the loan after the first month.

1B

6 Mr and Mrs Buckley borrow $130 000 to purchase a home. The interest rate is 8% p.a. and over a 20-year term the monthly repayment is $1087.37. a Copy and complete the table below. Month

Principal ($)

Interest ($)

Balance ($)

1

130 000.00

866.67

129 779.29

2

129 779.29

3 4 5 6 7 8 9 10 11 12

Chapter 1 Credit and borrowing

37

b Mr and Mrs Buckley decide to increase their monthly payment to $1500. Complete the table below. Month

Principal ($)

Interest ($)

Balance ($)

1

130 000.00

866.67

129 366.67

2

129 366.67

3 4 5 6 7 8 9 10 11 12 c How much less do Mr and Mrs Buckley owe at the end of one year by increasing their monthly repayment? 7 Mr and Mrs Stone borrow $225 000 for their home. The interest rate is 9.6% p.a. and the term of the loan is 25 years. The monthly repayment is $1989.48. a Calculate the total repayments made on this loan. b If Mr and Mrs Stone increase their monthly payments to $2000, the loan will be repaid in 24 years and 1 month. Calculate the amount they will save in repayments with this increase. ( 1 + r )n – 1 8 Use the formula E = ---------------------------- to calculate the effective interest rate on each of the n following flat rate loans (answer correct to 2 decimal places). a $4000 at 7% p.a. over 2 years b $12 000 at 11% p.a. over 5 years c $1320 at 23% p.a. over 2 years d $45 000 at 9.2% p.a. over 10 years 9 Yu-Ping borrows $13 500 for a holiday to Africa at 12.5% p.a. reducible interest over a 5-year term. The monthly repayments on the loan are $303.72. a Calculate the total repayments on the loan. b Calculate the amount of interest that Yu-Ping pays on the loan. c Calculate the equivalent flat rate of interest on the loan.

1B

1C

1C

38

Maths Quest General Mathematics HSC Course

1C

10 Kristen and Adrian borrow $150 000 for their home. They have the choice of two loans. Loan 1: At 8% p.a. interest over 25 years with fixed repayments of $1157.72. Loan 2: At 8.25% p.a interest over 25 years with minimum repayments of $1182.68 and an $8 per month account management fee. Kristen and Adrian believe they can afford to pay $1500 per month. If they do, Loan 2 will be repaid in 14 years and 2 months. Which loan should Kristen and Adrian choose if they can afford to pay the extra each month?

1C

11 Stephanie has a credit card with an outstanding balance of $423. Calculate the minimum payment that must be made if she must pay 5% of the balance, or $10, whichever is greater.

1D

12 Lorenzo has a credit card with an outstanding balance of $850. If the interest rate is 24% p.a., calculate the amount of interest that Lorenzo will be charged if the balance is not paid by the due date.

1D

13 Jessica pays for her car repairs, which total $256.50, using her credit card. The credit card has an interest rate of 15% p.a. and interest is charged from the date of purchase. Calculate the amount of interest charged after one month on this card.

1D

14 Study the extract from the credit card statement below. Interest rate = 19.5% p.a. Daily rate = ______ Date

Credit ($)

Debit ($)

1 Jan. 6 Jan.

2584.75 600 – repayment

15 Jan.

39.99 – purchase

1 Feb. 8 Feb. 15 Feb. 1 March

Balance ($)

??? – interest ??? – repayment 425.85 – purchase ??? – interest

a Calculate the daily rate of interest. (Take 1 year = 365.25 days and answer correct to 4 decimal places.) b Calculate the interest due for January. c If the minimum monthly payment of 5% of the outstanding balance is made on 8 February, calculate the amount of this repayment. d Calculate the interest for February.

1E

15 Use the table of repayments on page 30 to calculate the monthly repayment on each of the following loans. a $25 000 at 9% p.a. over a 10-year term b $45 000 at 14% p.a. over a 15-year term c $164 750 at 15% p.a. over a 25-year term d $425 000 at 12% p.a. over a 15-year term

1E

16 Mr and Mrs Rowe take out a $233 000 home loan at 12% p.a. over a 25-year term. a Use the table of repayments to calculate the amount of each monthly repayment. b After 3 years the balance on the loan has been reduced to $227 657. The interest rate then rises to 13% p.a. Calculate the new monthly repayment required to complete the loan within the existing term.

Chapter 1 Credit and borrowing

39

Practice examination questions 1 multiple choice The total repayments for a $3400 loan on a flat rate interest of 8.5% p.a. over a 3-year period are: A $867 B $942.78 C $4267 D 4342.78 2 multiple choice A $115 000 loan is repaid over a 25-year term at the rate of $1211.21 per month. The total amount of interest that is paid on this loan is: A $30 280.25 B $145 280.25 C $248 363.00 D $363 363.00 3 multiple choice A $150 000 loan is to be taken out. Which of the following loans will have the lowest total cost? A 4% p.a. flat rate interest to be repaid over 20 years B 8% p.a. reducible interest to be repaid over 20 years at $1254.66 per month C 9% p.a. reducible interest to be repaid over 15 years at $1521.40 per month D 8.5% p.a. reducible interest to be repaid over 15 years at $1512.49 per month with a $900 loan application fee and $12 per month account management fee 4 multiple choice Look at the table of loan repayments per $1000 shown below. Interest rate (p.a.) Term

9%

10%

11%

12%

10

$12.67

$13.22

$13.78

$14.35

15

$10.14

$10.75

$11.37

$12.00

20

$9.00

$9.65

$10.32

$11.01

25

$8.39

$9.09

$9.80

$10.53

Daniel has an $80 000 mortgage at 10% p.a. over 10 years. After interest rates rise to 12% Daniel extends the term of his loan to 15 years. What is the change in Daniel’s monthly repayments? A They increase by $1.13 per month. B They decrease by $1.22 per month. C They increase by $90.40 per month. D They decrease by $97.60 per month. 5 David buys a computer that has a cash price of $4600. David pays 10% deposit with the balance in weekly instalments at 13% p.a. flat rate interest over a period of 4 years. a Calculate the balance owing after David has paid the deposit. b Calculate the total repayments that David must make on this loan. c Calculate the amount of each weekly instalment ( 1 + r )n – 1 d Use the formula E = ---------------------------- to calculate the equivalent reducible interest rate on this n loan.

40

Maths Quest General Mathematics HSC Course

6 Mr and Mrs Tarrant borrow $186 500 to purchase a home. The interest rate is 9% p.a. and the loan is over a 20-year term. a Use the table below to calculate the amount of each monthly repayment. Interest rate (p.a.) Term

9%

10%

11%

12%

10

$12.67

$13.22

$13.78

$14.35

15

$10.14

$10.75

$11.37

$12.00

20

$9.00

$9.65

$10.32

$11.01

25

$8.39

$9.09

$9.80

$10.53

b Calculate the total amount that they can expect to make in repayments. c After 10 years the outstanding balance is $132 463 and the interest rate is increased to 11%. Calculate the amount of the monthly repayment they will need to make to complete the loan within the term. d The loan has a $5 per month account management fee. The Tarrants also had a $400 loan application fee and $132.75 in stamp duty to pay in establishing the loan. Calculate the total cost of the loan after 20 years.

CHAPTER

test yourself

1

7 Paul has a credit card that has an interest-free period. The interest rate is 21% p.a. a If Paul has an outstanding balance of $275.50, calculate the minimum payment he must make by the due date if it is 5% of the balance, or $10, whichever is greater. b If Paul pays only the minimum balance by the due date, calculate the balance owing for the next month. c Calculate the interest that Paul will be charged on his next month’s statement. d If Paul pays the whole balance off next month, is this card cheaper than a card without an interest-free period but an interest rate of 15% p.a.? Use calculations to justify your answer.

2

Further applications of area and volume

syllabus reference Measurement 5 • Further applications of area and volume

In this chapter 2A 2B 2C 2D

Area of parts of the circle Area of composite shapes Simpson’s rule Surface area of cylinders and spheres 2E Volume of composite solids 2F Error in measurement

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

2.1

2.2

Area of a circle

1 Find the area of a circle with: a radius 4 cm b radius 19.6 cm

c diameter 9 cm

d diameter 19.7 cm

Areas of squares, rectangles and triangles

2 Find the area of each of the following. 10.9 m a b

c 7.6 m

3.7 m

13.8 m 4.5 cm

2.6

Volume of cubes and rectangular prisms (3a, 3b); Volume of triangular prisms (3c)

3 Find the volume of: a

b

c 11 cm

2.7

8 cm 9 cm 24 cm

2.8

6 cm 26 cm

18 cm Volume of cylinders (4a); Volume of a sphere (4b)

4 Find the volume of: a 19 cm

2.9

b 12 m

8 cm

Volume of a pyramid

2.10

5

Find the volume of: 10 cm

7 cm

2.11

Error in linear measurement

6

For each of the following linear measurements, state the limits between which the true limits actually lie. a 15 cm (measured correct to the nearest centimetre) b 8.3 m (measured correct to 1 decimal place) c 4800 km (measured correct to the nearest 100 km)

Chapter 2 Further applications of area and volume

43

Area of parts of the circle From previous work you should know that the area of a circle can be calculated using the formula: A = r2

WORKED Example 1

Calculate the area of a circle with a radius of 7.2 cm. Give your answer correct to 2 decimal places. THINK 1 2 3

WRITE

Write the formula. Substitute for the radius. Calculate the area.

A = r2 A= (7.2)2 A = 162.86 cm2

A sector is the part of a circle between two radii as shown on the right. To calculate the area of a sector we find the fraction of the circle formed by the sector. For example, a semicircle is half of a circle and so the area of a semicircle is half the area of a full circle. A quadrant is a quarter of a circle and so the area is quarter that of a full circle. For other sectors the area is calculated by using the angle between the radii as a fraction of 360° and then multiplying by the area of the full circle. This can be written using the formula: A = --------- r 2 360 where

is the angle between the two radii.

WORKED Example 2

Calculate the area of the sector drawn on the right. Give your answer correct to 1 decimal place. 5 cm 80°

THINK

WRITE

1

Write the formula.

A=

2

Substitute for

A=

3

Calculate the area.

and r.

--------360 80 --------360

r2

A = 17.5 cm2

52

44

Maths Quest General Mathematics HSC Course

An annulus is the area between two circles that have the same centre (i.e. concentric circles). The area of an annulus is found by subtracting the area of the smaller circle from the area of the larger circle. This translates to the formula A = (R2 – r2) , where R is the radius of the outer circle and r is the radius of the inner circle.

WORKED Example 3 Calculate the area of the annulus on the right. Give your answer correct to 1 decimal place.

5.7 cm 3.2 cm

THINK

WRITE

1

Write the formula.

A = (R2 – r 2)

2

Substitute R = 5.7 and r = 3.2.

A = (5.72 – 3.22)

3

Calculate.

A = 69.9 cm2

An ellipse is an oval shape and therefore does not have a constant radius. The greatest distance from the centre of the ellipse to the circumference is called the semi-major axis, a, while the smallest distance is called the semi-minor axis, b, as shown in the figure on the right. The area of an ellipse is calculated using the formula, found on the formula sheet:

semi-minor axis (b) semi-major axis (a)

A = ab

WORKED Example 4 Calculate the area of the ellipse drawn on the right. Give your answer correct to 2 decimal places.

4.2 m 6.6 m

THINK

WRITE

1

Write the formula.

A = ab

2

Substitute for the values of a and b.

A=

3

Calculate the area.

6.6

A = 87.08 m

4.2 2

Chapter 2 Further applications of area and volume

45

remember

1. The area of a circle is found using the formula A = r2. 2. The area of a sector can be found by multiplying the area of a full circle by the fraction of the circle given by the angle in the sector. You can use the formula A = --------r 2. 360 3. An annulus is the area between two concentric circles. The area is found by using the formula A = (R2 – r2) , where R is the radius of the outer circle and r is the radius of the inner circle. 4. An ellipse is an oval shape. The area is calculated using the formula A = ab , where a is the length of the semi-major axis and b is the length of the semi-minor axis.

2A WORKED

Example

1 Calculate the area of the circle drawn on the right, correct to 1 decimal place.

2.1 6.4 cm

Area of a circle

2 Calculate the area of each of the circles drawn below, correct to 2 decimal places. a b c 33 mm 9 cm

d

7.4 m

e

26.5 cm

f 6.02 m 3.84 m

3 Calculate the area of a circle that has a diameter of 15 m. Give your answer correct to 1 decimal place. WORKED

Example

2

SkillS

HEET

1

Area of parts of the circle

4 Calculate the area of the sector drawn on the right. Give your answer correct to 1 decimal place. 7.2 m

46

Maths Quest General Mathematics HSC Course

5 Calculate the area of each of the sectors drawn below. Give each answer correct to 2 decimal places. a b c

23 m

5.2 cm

135° 74 mm

60°

20°

d

e

f

9.2 mm 150°

39 mm 240° 19.5 m

72°

6 Calculate, correct to 1 decimal place, the area of a semicircle with a diameter of 45.9 cm. WORKED

Example

3

7 Calculate the area of the annulus shown at right, correct to 1 decimal place.

12 cm

6 cm

8 Calculate the area of each annulus drawn below, correct to 3 significant figures. a b c 9.7 m

77 mm

20 cm 18 cm

13 mm

4.2 m

9 A circular garden of diameter 5 m is to have concrete laid around it. The concrete is to be 1 m wide. a What is the radius of the garden? b What is the radius of the concrete circle? c Calculate the area of the concrete, correct to 1 decimal place. 10 Calculate the area of the ellipse drawn on the right, correct to 4 1 decimal place.

WORKED

Example

6 cm 10 cm

Chapter 2 Further applications of area and volume

47

11 Calculate the area of each of the ellipses drawn below. Give each answer correct to the nearest whole number. a b c 7.2 m

34 mm

14 cm

13.6 m

56 mm

21 cm

12 multiple choice The area of a circle with a diameter of 4.8 m is closest to: A 15 m2 B 18 m2 C 36 m2

D 72 m2

13 multiple choice Which of the following calculations will give the area of the sector shown on the right?

45° 8m

A

1 --8

× π × 42

B

1 --8

× π × 82

C 1-4 × π × 4 2

D 1-4 × π × 8 2

14 multiple choice The area of the ellipse drawn on the right is closest to:

86 cm 1.2 m

A 32 400 cm2

B 324 m2

C 5900 cm2

D 59 m2

15 A circular area is pegged out and has a diameter of 10 m. a Calculate the area of this circle, correct to 1 decimal place. b A garden is to be dug which is 3 m wide around the area that has been pegged out. Calculate the area of the garden to be dug. Give your answer correct to 1 decimal place. c In the garden a sector with an angle of 75° at the centre is to be used to plant roses. Calculate the area of the rose garden, correct to 1 decimal place. 16 A circle has a diameter of 20 cm. a Calculate the area of this circle, correct to 2 decimal places. b An ellipse is drawn such that the radius of the circle forms the semi-major axis. The semi-minor axis is to have a length equal to half the radius of the circle. Calculate the length of the semi-minor axis. c Calculate the area of the ellipse, correct to 2 decimal places.

48

Maths Quest General Mathematics HSC Course

Area of composite shapes A composite shape is a shape that is made up of two or more regular shapes. The area of a composite shape is found by splitting the area into two or more regular shapes and calculating the area of each separately before adding them together. In many cases it will be necessary to calculate the length of a missing side before calculating the area. There will sometimes be more than one way to split the composite shape.

WORKED Example 5

6 cm

Find the area of the figure at right. 18 cm 10 cm 12 cm

THINK 1

WRITE

Copy the diagram and divide the shape into two rectangles.

6 cm A1 8 cm 18 cm 10 cm

A2 12 cm

3

Calculate the length of the missing side in rectangle 1. (Write this on the diagram.) Calculate the area of rectangle 1.

4

Calculate the area of rectangle 2.

5

Add together the two areas.

2

18

10 = 8 cm

A1 = 6 8 A1 = 48 cm2 A2 = 10 12 A1 = 120 cm2 Area = 48 + 120 Area = 168 cm2

Composite areas that involve triangles may require you to also make a calculation using Pythagoras’ theorem.

WORKED Example 6

13 m

Find the area of the figure on the right. 10 m 24 m

THINK 1

Draw the triangle at the top and cut the isosceles triangle in half.

WRITE 13 m a 12 m

Chapter 2 Further applications of area and volume

THINK

49

WRITE

2

Calculate the perpendicular height using Pythagoras’ theorem.

a2 = c2 b2 = 132 122 = 169 144 = 25 a = 25 =5m

3

Calculate the area of the triangle.

A=

1 --2

24

5

2

4

Calculate the area of the rectangle.

5

Add the two areas together.

= 60 m A = 24 10 = 240 m2 Area = 60 + 240 Area = 300 m2

Composite areas can also be calculated by using subtraction rather than addition. In these cases we calculate the larger area and subtract the smaller area in the same way as we did with annuluses in the previous section.

WORKED Example 7

Find the shaded area in the figure on the right.

6 cm

20 cm

30 cm

THINK

WRITE

1

Calculate the area of the rectangle.

2

Calculate the area of the circle.

3

Subtract the areas.

A = 30 20 A = 600 cm2 A= 62 A = 113.1 cm2 Area = 600 113.1 Area = 486.9 cm2

remember 1. To find the area of any composite figure, divide the shape into smaller regular shapes and calculate each area separately. 2. You may have to use Pythagoras’ theorem to find missing pieces of information. 3. Check if the best way to solve the question is by adding two areas or by subtracting one area from the other to find the remaining area.

50

Maths Quest General Mathematics HSC Course

SkillS

2B HEET

2.2

WORKED

Example

5 Areas of squares, rectangles and triangles

Area of composite shapes

1 Copy the figure on the right into your workbook and calculate its area by dividing it into two rectangles.

4m

18 m 11 m 20 m

2 Find the area of each of the figures below. Where necessary, give your answer correct to 1 decimal place. a

7 cm b

c

5 cm

18 cm 12 cm

19 cm

16 cm 25 cm

6 cm 40 cm

5 cm 22 cm

d

e

f

12 cm

4 cm

8 cm

8 cm

4 cm 4 cm

SkillS

16 cm

HEET

2.3 Using Pythagoras’ theorem

Cabr

omet i Ge ry

Pythagoras’ calculations

3 Look at the triangle on the right. a Use Pythagoras’ theorem to find the perpendicular height of the triangle. b Calculate the area of the triangle.

10 cm

6 cm

17 cm

15 cm

4 Below is an isosceles triangle.

8m

12 m

a Use Pythagoras’ theorem to find the perpendicular height of the triangle, correct to 1 decimal place. b Calculate the area of the triangle.

51

Chapter 2 Further applications of area and volume

E

sheet

5 Calculate the area of each of the triangles below. Where necessary, give your answer L Spre XCE ad correct to 1 decimal place. Pythagoras a b c 25 cm 26 m

24 m

Mensuration 124 mm

WORKED

Example

6

6 Find the area of each of the composite figures drawn below. a b c 25 mm 13 cm 17 m

52 mm

15 mm

48 mm 54 mm

13 m

12 cm 30 m

7 multiple choice The area of the composite figure on the right is closest to: A 139 m2 B 257 m2 2 C 314 m D 414 m2 10 m

8 multiple choice The area of the figure drawn on the right is: A 36 m2 B 54 m2 2 C 72 m D 144 m2

12 m

6m

9 A block of land is in the shape of a square with an equilateral triangle on top. Each side of the block of land is 50 m. a Draw a diagram of the block of land. b Find the perimeter of the block of land. c Find the area of the block of land. 10 In each of the following, find the area of the shaded region. Where necessary, give your answer correct to 1 decimal place. 7 a b c 12 m

WORKED

Example

4 cm

10 cm

9 cm 3 cm

16 cm

8m

9 cm

7. 1

1.9 m 7.4 m

36 mm 40 mm

10 cm 95 mm

3.1 m

f 112 mm

e

cm

d

am progr –C

asio

GC

48 cm

Maths Quest General Mathematics HSC Course

11 An athletics track consists of a rectangle with two semicircular ends. The dimensions are shown in the diagram on the right. The track is to have a synthetic running surface laid. Calculate the area which is to be laid with the running surface, correct to the nearest square metre.

70 m 90 m

12 A garden is to have a concrete path laid around it. The garden is rectangular in shape and measures 40 m by 25 m. The path around it is to be 1 m wide. a Draw a diagram of the garden and the path. b Calculate the area of the garden. c Calculate the area of the concrete that needs to be laid. d If the cost of laying concrete is $17.50 per m2, calculate the cost of laying the path.

1 Calculate the area of each of the figures drawn below. Where necessary, give your answer correct to 1 decimal place. 1

2

3 5.8 cm 9.4 cm

12 cm

6.3 m

4

5

6

10 cm

91 mm 30 cm

25 cm 62 mm

20 cm 4 cm 25 cm

24 cm

7

8

9 12 cm 76

mm

m

20 m

32

m

40 m 40 cm

10 6 cm

52

12

cm

82 m

15 cm

Chapter 2 Further applications of area and volume

53

Simpson’s rule Simpson’s rule is a method used to approximate the area of an irregular figure. Simpson’s rule approximates an area by taking a straight boundary and dividing the area into two strips. The height of each strip (h) is measured. Three measurements are then taken perpendicular to the straight boundary, as shown in the figure on the right. The formula for Simpson’s rule is: h A --- ( d f + 4d m + d l ) 3 where h = distance between successive measuements df = first measurement dm = middle measurement dl = last measurement.

dm df h

WORKED Example 8 Use Simpson’s rule to approximate the area shown on the right. 30 m 10 m 90 m

THINK 1

Calculate h.

2

Write down the values of df , dm and dl.

3

Write the formula.

4

Substitute.

5

Calculate.

dl

WRITE h = 90 ÷ 2 = 45 df = 10, dm = 30, dl = 0 h A --- ( d f + 4d m + d l ) 3 45 A ------ ( 10 + 4 30 + 0 ) 3 = 15 130 1950 m2

Could Simpson’s rule be used to estimate the areas of these irregular shapes from nature?

h

54

Maths Quest General Mathematics HSC Course

Simpson’s rule can be used to approximate an irregular area without a straight edge. This is done by constructing a line as in the diagram below and approximating the area of each section separately.

WORKED Example 9 Use Simpson’s rule to find an approximation for the area shown on the right.

30 m 30 m

30 m

10 m

17 m

THINK

WRITE

2

Write down the value of h. For the top area, write down the values of df , dm and dl .

3

Write the formula.

4

Substitute.

5

Calculate the top area.

6

For the bottom area, write down the values of df , dm and dl .

7

Write down the formula.

8

Substitute.

9

Calculate the bottom area.

10

Add the two areas together.

1

h = 30 df = 0, dm = 30, dl = 10 h --- ( d f + 4d m + d l ) 3 30 A ------ ( 0 + 4 30 + 10 ) 3 10 130 1300 m2 df = 0, dm = 17, dl = 0 A

h --- ( d f + 4d m + d l ) 3 30 A ------ ( 0 + 4 17 + 0 ) 3 10 68 680 m2 Area 1300 + 680 Area 1980 m2 A

Simpson’s rule approximates an area, it does not give an exact measurement. To obtain a better approximation, Simpson’s rule can be applied several times to the area. This is done by splitting the area in half and applying Simpson’s rule separately to each half.

30 m

29 m

24 m

31 m

Use two applications of Simpson’s rule to approximate the area on the right.

32 m

WORKED Example 10 105 m

THINK 1 2

Calculate h by dividing 105 by 4. (We are using 4 sub-intervals.) Apply Simpson’s rule to the left half. Write the values of df , dm and dl .

WRITE h = 105 ÷ 4 = 26.25 df = 32, dm = 31, dl = 24

Chapter 2 Further applications of area and volume

THINK

WRITE

3

Write the formula.

4

Substitute.

5

Calculate the approximate area of the left half. Apply Simpson’s rule to the left half. Write the values of df , dm and dl .

6

h --- ( d f + 4d m + d l ) 3 26.25 A ------------- ( 32 + 4 31 + 24 ) 3 8.75 180 1575 m2 df = 24, dm = 29, dl = 30 A

7

Write the formula.

8

Substitute.

9

Calculate the approximate area of the right half. Add the areas together.

10

55

h --- ( d f + 4d m + d l ) 3 26.25 A ------------- ( 24 + 4 29 + 30 ) 3 8.75 170 1487.5 m2 Area 1575 + 1487.5 Area 3062.5 m2 A

remember 1. Simpson’s rule is a method of approximating irregular areas. h 2. The Simpson’s rule formula is A --- ( d f + 4d m + d l ) , where h is the 3 distance between successive measurements, df is the first measurement, dm is the middle measurement and dl is the last measurement. 3. A better approximation of an area can be found by using Simpson’s rule twice.

2.4

60 m

18 m

8

1 The diagram on the right is of a part of a creek. a State the value of h. b State the value of df , dm and dl . c Use Simpson’s rule to approximate the area of this section of the creek.

9m

Example

Substitution into formulas

Simpson’s rule

40 m

WORKED

HEET

2C

SkillS

56

Maths Quest General Mathematics HSC Course

72 m

40 m

16 m

0m

12 m

28 m

6m

12 m

10 m

35 m

2 Use Simpson’s rule to approximate each of the areas below. a b c

54 m

48 m A1

30 m

30 m

5 m 18 m

9

3 The irregular area on the right has been divided into two areas labelled A1 (upper area) and A2 (lower area). a Use Simpson’s rule to find an approximation for Al . b Use Simpson’s rule to find an approximation for A2. c What is the approximate total area of the figure?

19 m 11 m

Example

7m

WORKED

A2

21 m

31 m 27 m

7m

51 m

22 m

5 multiple choice Consider the figure drawn on the right. Simpson’s rule gives an approximate area of: A 1200 m2 B 2400 m2 2 C 3495 m D 6990 m2

27 m 40 m

16 m

21 m

27 m

16 m

12 m 10 m

0

23 m

45 m

14 m 6 m

17 m

45 m 12 m

22 m 11 m

4 Use Simpson’s rule to find an approximation for each of the areas below. a b c

90 m

6 multiple choice If we apply Simpson’s rule twice, how many measurements from the traverse line need to be taken? A4 B 5 C7 D9

25 m

36 m

10 m

45 m

50 m

18 m 18 m 18 m 18 m

10 m 10 m 10 m 10 m

60 m 60 m 60 m 60 m

22 m

33 m

11 m

50 m

44 m

20 m

71 m

42 m

87 m

102 m

8 Use Simpson’s rule twice to approximate each of the areas drawn below. a b c

45 m

10

7 Use Simpson’s rule twice to approximate the area on the right.

63 m

Example

54 m

WORKED

21 m 21 m 21 m 21 m

Chapter 2 Further applications of area and volume

36 m

27 m

32 m 15 m

9 The figure on the right is of a cross-section of a waterway. a Use Simpson’s rule once to find an approximate area of this section of land. b Use Simpson’s rule twice to obtain a better approximation for this area of land.

57

Work

20 m 30 m 35 m 36 m 38 m 41 m 45 m 30 m 24 m

10 Apply Simpson’s rule four times to approximate the area on the right.

9m 9m 9m 9m 9m 9m 9m 9m

Surface area of cylinders and spheres From earlier work you should remember that surface area is the area of all surfaces of a 3-dimensional shape. Consider a closed cylinder with a radius (r) and a perpendicular height (h). The surface of the cylinder h consists of two circles and a rectangle. Area of top = r 2 r Area of bottom = r 2 The rectangular side of the cylinder will have a length equal to the circumference of the circle (2 r) and a width equal to the height (h) of the cylinder. Area of side = 2 rh The surface area of the closed cylinder can be calculated using the formula: SA = 2 r 2 + 2 rh

WORKED Example 11

Calculate the surface area of the closed cylinder drawn on the right. Give your answer correct to 1 decimal place. 10 cm 9 cm

THINK 1 2 3

WRITE

Write the formula. Substitute the values of r and h. Calculate the surface area.

SA = 2 r 2 + 2 rh SA = 2 92 + 2 SA = 1074.4 cm2

9

10

For cylinders, before calculating the surface area you need to consider whether the cylinder is open or closed. In the case of an open cylinder there is no top and so the formula needs to be written as: SA = r 2 + 2 rh

T SHEE

2.1

58

Maths Quest General Mathematics HSC Course

Note: On the formula sheet in the exam only the formula for the closed cylinder is provided. You will need to adapt the formula yourself for examples such as this.

WORKED Example 12 Calculate the surface area of an open cylinder with a radius of 6.5 cm and a height of 10.8 cm. Give your answer correct to 1 decimal place. THINK 1 2 3

WRITE

Write the formula. Substitute the values of r and h. Calculate the surface area.

SA = r 2 + 2 rh SA = (6.5)2 + 2 SA = 573.8 cm2

A sphere is a round 3-dimensional shape, and the only measurement given is the radius (r). The surface area of a sphere can be calculated using the formula: SA = 4 r 2

WORKED Example 13 Calculate the surface area of the sphere drawn on the right. Give the answer correct to 1 decimal place.

THINK 1 2 3

2.7 cm

WRITE

Write the formula. Substitute the value of r. Calculate the surface area.

SA = 4 r 2 SA = 4 (2.7)2 2 SA = 91.6 cm

remember 1. The surface area of a closed cylinder is found using the formula SA = 2 r 2 + 2 rh . 2. If the cylinder is an open cylinder, the surface area formula becomes SA = r 2 + 2 rh. 3. The surface area of a sphere is found using the formula SA = 4 r 2 .

The Atomium, Brussels

6.5

10.8

r

59

Chapter 2 Further applications of area and volume

Surface area of cylinders and spheres

2D

2.5

Example

11

1 Calculate the surface area of a closed cylinder with a radius of 5 cm and a height of 11 cm. Give your answer correct to 1 decimal place. Circumference of a circle

12 cm

5 cm

1.6 m

2 Calculate the surface area of each of the closed cylinders drawn below. Give each answer correct to 1 decimal place. a b c 20 cm

1.1 m 3 cm

d

e

f 5.9 cm

20 cm

1.5 m

5.9 cm r

2.3 m

r = 5 cm

3 Calculate the surface area of a closed cylinder with a diameter of 3.4 m and a height of 1.8 m. Give your answer correct to 1 decimal place. Example

12

4 Calculate the surface area of an open cylinder with a radius of 4 cm and a height of 16 cm. Give your answer correct to the nearest whole number. 5 Calculate the surface area of each of the following open cylinders. Give each answer correct to 1 decimal place. a b c

30 cm

22 cm

13.3 cm 9.6 cm r

20 cm

r = 4.1 cm

d

e

f 50 cm

3.2 m

WORKED

23.2 cm 4m

2.4 cm 4 cm

6 A can of fruit is made of stainless steel. The can has a radius of 3.5 cm and a height of 7 cm. A label is to be wrapped around the can. a Calculate the amount of steel needed to make the can (correct to the nearest whole number). b Calculate the area of the label (correct to the nearest whole number).

SkillS HEET

WORKED

60

Maths Quest General Mathematics HSC Course

WORKED

Example

13

7 Calculate the surface area of a sphere with a radius of 3 cm. Give your answer correct to the nearest whole number. 8 Calculate the surface area of each of the spheres drawn below. Give each answer correct to 1 decimal place. a b c 2.1 cm 14 cm

8 cm

d

e 1m

f 3.4 cm

1.8 m

9 Calculate the surface area of a sphere with a diameter of 42 cm. Give your answer correct to the nearest whole number. 10 multiple choice An open cylinder has a diameter of 12 cm and a height of 15 cm. Which of the following calculations gives the correct surface area of the cylinder? A 62 + 2 6 15 B 2 62 + 2 6 15 2 C 12 + 2 12 15 D2 122 + 2 12 15 11 multiple choice Which of the following figures has the greatest surface area? A A closed cylinder with a radius of 5 cm and a height of 10 cm B An open cylinder with a radius of 6 cm and a height of 10 cm C A cylinder open at both ends with a radius of 7 cm and a height of 10 cm D A sphere with a radius of 6 cm 12 An open cylinder has a diameter and height of 12 cm. a Calculate the surface area of the cylinder (correct to the nearest whole number). b A sphere sits exactly inside this cylinder. Calculate the surface area of this sphere (correct to the nearest whole number). 13 A cylindrical can is to contain three tennis balls each having a diameter of 6 cm. a Calculate the surface area of each ball. b The three balls fit exactly inside the can. State the radius and height of the can. c The can is open and made of stainless steel, except the top which will be plastic. Calculate the area of the plastic lid (correct to the nearest whole number). d Calculate the amount of stainless steel in the can (correct to the nearest whole number). e Calculate the area of a paper label that is to be wrapped around the can (correct to the nearest whole number).

Chapter 2 Further applications of area and volume

61

Computer Application 1 Minimising surface area E

1. In cell B3 enter the volume of the cylinder, 1000. 2. In cell A6 enter a radius of 1. In cell A7 enter a radius of 2 and so on up to a radius of 20. 3. The formula that has been entered in cell B6 will give the height of the cylinder corresponding to the radius for the given volume. 4. The surface area of each possible cylinder is in column D. Use the charting function on the spreadsheet to graph the surface area against the radius. 5. What are the most cost-efficient dimensions of the drink container?

Challenge exercise Use one of the other worksheets to find the most efficient dimensions to make a rectangular prism of volume 1000 cm3 and a cone of volume 200 cm3.

sheet

L Spre XCE ad

Access the spreadsheet Volume from the Maths Quest General Mathematics HSC Volume Course CD-ROM. A cylindrical drink container is to have a capacity of 1 litre (volume = 1000 cm3). We are going to calculate the most cost efficient dimensions to make the container. To do this, we want to make the container with as little material as possible, in other words we want to minimise the surface area of the cylinder. The spreadsheet should look as shown below.

62

Maths Quest General Mathematics HSC Course

Packaging A company makes tennis balls that have a diameter of 6.5 cm. The tennis balls are to be sold in packs of four. 1 Calculate the surface area of the packaging needed if the balls are packed in a cylindrical tube that just fits all four balls as shown on the right.

2 Calculate the amount of packaging needed if the balls are packed in a rectangular prism.

3 Calculate the amount of packaging needed if the balls are packed in a 2 2 design as shown on the right.

4 Design the most effective way of packaging nine tennis balls.

Volume of composite solids Many solid shapes are a composition of two or more regular solids. To calculate the volume of such a figure, we need to determine the best method for each particular part. Many irregular shapes may still be prisms. A prism is a shape in which every cross-section taken parallel to the base shape is equal to that base shape. The formula for the volume of a prism is: V = Ah where A is the area of the base shape and h is the height. Remember that the base of the prism is not necessarily the bottom. The base is the shape that is constant throughout the prism and will usually be drawn as the front of the prism. This means that the height will be drawn perpendicular to the base. To calculate the volume of any prism, we first calculate the area of the base and then multiply by the height.

63

Chapter 2 Further applications of area and volume

WORKED Example 14 Find the volume of the figure drawn on the right.

6 cm

12 cm

4 cm

10 cm

Divide the front face into two rectangles.

4 cm 12 cm

1

WRITE

A1 A2

6 cm

THINK

3 cm

10 cm 2

Calculate the area of each.

3

Add the areas together to find the value of A. Write the formula. Substitute A = 84 and h = 3. Calculate.

4 5 6

A1 = 4 12 = 48 cm2 A = 48 + 36 = 84 cm2 V=A h = 84 3 = 252 cm3

A2 = 6 6 = 36 cm2

If the shape is not a prism, you may need to divide it into two or more regular 3dimensional shapes. You could then calculate the volume by finding the volume of each shape separately. You will need to use important volume formulas that appear on the formula sheet: Cone: V = 1--- r 2h Cylinder: V = r 2h Pyramid: V = 1--- Ah Sphere: V = 4--- r 3 3

3

WORKED Example 15 Calculate the volume of the figure drawn on the right, correct to 2 decimal places. 2.4 cm 1.2 cm

THINK 1 2 3 4

The shape is a cylinder with a hemisphere on top. Write down the formula for the volume of a cylinder. Substitute r = 1.2 and h = 2.4. Calculate the volume of the cylinder.

WRITE

V = r 2h V= (1.2)2 2.4 V = 10.857 cm3 Continued over page

3

64

Maths Quest General Mathematics HSC Course

THINK

WRITE r3 ÷ 2

5

Write down the formula for the volume of a hemisphere. (This is the formula for the volume of a sphere divided by 2.)

V=

4 --3

6

Substitute r = 1.2.

V=

4 --3

7

Calculate the volume of the hemisphere.

V = 3.619 cm3

8

Add the two volumes together.

Volume = 10.857 + 3.619 Volume = 14.48 cm3

(1.2)3 ÷ 2

In many cases a volume question may be presented in the form of a practical problem.

WORKED Example 16 A water storage tank is in the shape of a cube of side length 1.8 m, surmounted by a cylinder of diameter 1 m with a height of 0.5 m. Calculate the capacity of the tank, correct to the nearest 100 litres. THINK 1

WRITE

Draw a diagram of the water tank.

0.5 m 1m

1.8 m 2

Calculate the volume of the cube using the formula V = s3.

V = s3 V = 1.83 V = 5.832 m3

3

Calculate the volume of the cylinder using the formula V = r 2h.

V = r 2h V= 0.52 0.5 V = 0.393 m3

4

Add the volumes together.

Volume = 5.832 + 0.393 Volume = 6.225 m3

5

Calculate the capacity of the tank using 1 m3 = 1000 L.

Capacity = 6.225 1000 Capacity = 6225 L

6

Give an answer in words.

The capacity of the tank is approximately 6200 litres.

65

Chapter 2 Further applications of area and volume

remember 1. To find the volume of any prism, use the formula V = A ¥ h, where A is the area of the base and h is the height. 2. Important volume formulas: Cone: V = 1--- p r 2h Cylinder: V = p r 2h 3 1 Pyramid: V = --- Ah Sphere: V = 4--- p r 3 3 3 where r = radius, h = perpendicular height, A = area of base 3. For other shapes, calculate the volume of each part of the shape separately, then add together each part at the end. 4. Remember to begin a worded or problem question with a diagram and finish with a word answer.

2E

Volume of composite solids 6 cm

2.6 Volume of cubes and rectangular prisms

4 cm 20 cm

12 cm

5 cm

40 cm

e

0.7 m

2.3 m 0.4 m

0.6 m 1m

5m

1.5 m 2m

2.1 m

WORKED

Example

3 Consider the figure on the right. The shape consists of a cube with a square pyramid on top. a What is the volume of the cube? b What is the volume of the square pyramid? c What is the total volume of this figure?

1.5 m

2.7

SkillS HEET

15

3 cm

f 4 m 0.5 m

d

20 cm

12 cm 25 cm

15 cm

4 cm 20 cm

10 cm

2 Calculate the volume of each of the figures drawn below. a b c 5 cm 12 cm

SkillS HEET

14

1 Look at the figure drawn on the right. a Find the area of the front face. b Use the formula V = A ¥ h to calculate the volume of the prism.

5 cm

Example

18 cm

WORKED

Volume of triangular prisms

2m

66

Maths Quest General Mathematics HSC Course

4 The figure on the right is a cylinder with a cone mounted on top. a Calculate the volume of the cylinder, correct to the nearest cm3. b Calculate the volume of the cone, correct to the nearest cm3. c What is the total volume of the figure?

40 cm

50 cm

SkillS

12 cm

HEET

2.8

5 Calculate the volume of each of the figures drawn below, correct to 1 decimal place. a b c

Volume of a cylinder

3 cm 34 cm 5 cm

SkillS

r

HEET

2.9 Volume of a sphere

r =12 cm

50 cm

6 multiple choice Which of the figures drawn below is not a prism? A B

C

D

7 multiple choice The volume of the figure drawn on the right is closest to: A 718 cm3 B 1437 cm3 3 C 2155 cm D 2873 cm3 8 A fish tank is in the shape of a rectangular prism. The base measures 45 cm by 25 cm. The tank is filled to a depth of 15 cm. a Calculate the volume of water in the tank in cm3. b Given that 1 cm3 = 1 mL calculate, in litres, the amount of water in the tank.

14 cm 7 cm

67

Chapter 2 Further applications of area and volume

Example

16

9 A hemispherical wine glass of radius 2.5 cm is joined to a cylinder of radius 1 cm and height 5 cm. The glass then rests on a solid base. a Draw a diagram of the wine glass. b Calculate the capacity of the glass, to the nearest 10 mL. c How many glasses of wine can be poured from a 1 litre bottle? 10 The figure on the right is the cross-section of a concrete pipe used as a sewage outlet. a Calculate the area of a cross-section of the pipe, correct to 2 decimal places. b Calculate the amount of concrete needed to make a 10 m length of this pipe.

3m 2.5 m

11 A commemorative cricket ball has a diameter of 7 cm. It is to be preserved in a cubic case that will allow 5 mm on each side of the ball. a What will the side length of the cubic case be? b Calculate the amount of empty space inside the case, to the nearest whole number. c Calculate the percentage of space inside the case occupied by the ball, to the nearest whole number.

6 mm

Maximising volume You have been given a piece of sheet metal that is in the shape of a square with a side length of 2 m. The corners are to be cut and 9 cm the sides bent upwards to form a rectangular prism, as shown in the figure on the right. 3 cm 1 If a square of side length 1 cm is cut from each corner, what will be the length and width of the rectangular prism? 2 What will be the volume of this rectangular prism? 3 What will be the volume of the prism if a square of side length 2 cm is cut from each corner? 4 Find the size of the square to be cut from each corner that will make a prism of maximum volume. This exercise can be modelled using a spreadsheet or a graphics calculator.

2.10 SkillS HEET

12 A diamond is cut into the shape of two square-based pyramids as shown on the right. Each mm3 of the diamond has a mass of 0.04 g. Calculate the mass of the diamond.

6 mm

WORKED

Volume of a pyramid

Maths Quest General Mathematics HSC Course

2 1 Calculate the area of a circle with a diameter of 8.6 cm, correct to 1 decimal place.

2 Calculate the area of the annulus shown below, correct to 2 decimal places. 9 cm 3 cm

4 Calculate the area of the figure below. 10 cm

9 cm

29 cm

3 Calculate the area of the sector below, correct to 1 decimal place.

13.2 cm 28 cm

85°

5 Calculate the shaded area in the figure drawn below, correct to 2 decimal places.

6 Use Simpson’s rule to approximate the area shown below. 43 m

70 m 21 m 4.6 cm 70 m 9.7 cm 32 m

7 Calculate the surface area of a closed cylinder with a radius of 10 cm and a height of 23 cm. Give your answer correct to the nearest whole number. 9 Calculate the volume of the prism drawn below.

8 Calculate the surface area of a sphere with a radius of 1.3 m. Give your answer correct to 3 decimal places. 10 Calculate the volume of the solid below, correct to the nearest whole number. 4 cm

20.3 cm

68

13.4 cm

9.1 cm 13.7 cm

8 cm

Chapter 2 Further applications of area and volume

69

Error in measurement As we saw in the preliminary course, all measurements are approximations. The degree of accuracy in any measurement is restricted by the accuracy of the measuring device and the degree of practicality. We have previously seen that the maximum error in any measurement is half of the smallest unit of measurement. This error is compounded when further calculations such as surface area or volume are made.

In the rectangular prism on the right, the length, breadth and height have been measured, correct to the nearest centimetre. a Calculate the volume of the rectangular prism. b Calculate the greatest possible error in the volume.

8 cm

WORKED Example 17

15 cm 20 cm

THINK

WRITE

a Calculate the volume of the rectangular prism.

a V=l w h = 20 15 8 = 2400 cm3

b

b Smallest possible dimensions: l = 19.5, w = 14.5, h = 7.5 V=l w h = 19.5 14.5 7.5 = 2120.625 cm3 Largest possible dimensions: l = 20.5, w = 15.5, h = 8.5 V=l w h = 20.5 15.5 8.5 = 2700.875 cm3 Maximum error = 2700.875 2400 Maximum error = 300.875 cm3

1

2

3

4

5

Write the smallest possible dimensions of the prism. Calculate the smallest possible volume. Write the largest possible dimensions of the prism. Calculate the largest possible volume. Calculate the maximum error.

As can be seen in the above example, a possible error of 0.5 cm in the linear measurement compounds to an error of 300.875 cm3 in the volume measurement. Mismeasurements that are made will compound all further calculations.

70

Maths Quest General Mathematics HSC Course

WORKED Example 18 A swimming pool is built in the shape of a rectangular prism with a length of 10.2 m, a width of 7.5 m and a depth of 1.5 m. The floor and the sides of the pool need to be cemented. a Calculate the area that is to be cemented. b The concreter mismeasured the length of the pool as 9.4 m. Calculate the error in the area calculation. c Calculate the percentage error (correct to 1 decimal place) in the area calculation. THINK

WRITE

a

a Area of floor = 10.2 7.5 Area of floor = 76.5 m2 Area of ends = 7.5 1.5 Area of ends = 11.25 m2 Area of sides = 10.2 1.5 Area of sides = 15.3 m2 Total area = 76.5 + 2 11.25 + 2 Total area = 129.6 m2

b

1

Calculate the area of the pool floor.

2

Calculate the area of the ends.

3

Calculate the area of the sides.

4

Calculate the total area to be cemented.

1

Use the incorrect measurement to repeat all the above calculations.

2

Find the difference between the two answers.

c Write the error as a percentage of the correct answer.

b Area of floor = 9.4 7.5 Area of floor = 70.5 m2 Area of ends = 7.5 1.5 Area of ends = 11.25 m2 Area of sides = 9.4 1.5 Area of sides = 14.1 m2 Total area = 70.5 + 2 11.25 + 2 Total area = 121.2 m2 Error = 129.6 121.2 Error = 8.4 m2 8.4 c Percentage error = ------------129.6 Percentage error = 6.5%

15.3

14.1

100%

remember 1. All measurements are approximations. The accuracy of any measurement is limited by the instrument used and the most practical degree of accuracy. 2. The maximum error in any linear measurement is half the smallest unit used. 3. Any error made in linear measurement will compound when used in further calculations such as those for surface area or volume.

Chapter 2 Further applications of area and volume

WORKED

Example

1 In the figure on the right each measurement has been taken to the nearest centimetre. a Calculate the volume of the figure. b Calculate the maximum error in the volume calculation.

2.11 SkillS HEET

17

Error in measurement 12 cm

2F

71

6 cm 16 cm

Error in linear measurement

2 The radius of a circle is measured as 7.6 cm, correct to 1 decimal place. a What is the maximum possible error in the measurement of the radius? b Calculate the area of the circle. Give your answer correct to 1 decimal place. c Calculate the maximum possible error in the area of the circle. d Calculate the maximum possible error in the area of the circle as a percentage of the area. 3 A cube has a side length of 16 mm, correct to the nearest millimetre. a Calculate the volume of the cube. b Calculate the smallest possible volume of the cube. c Calculate the largest possible volume of the cube. d Calculate the maximum possible percentage error in the volume of the cube. e Calculate the surface area of the cube. f Calculate the smallest possible surface area of the cube. g Calculate the largest possible surface area of the cube. h Calculate the maximum possible percentage error in the surface area of the cube. 4 A cylinder has a radius of 4 cm and a height of 6 cm with each measurement being taken correct to the nearest centimetre. a Calculate the volume of the cylinder (correct to the nearest whole number). b Calculate the smallest possible volume of the cylinder (correct to the nearest whole number). c Calculate the largest possible volume of the cylinder (correct to the nearest whole number). d Calculate the greatest possible percentage error in the volume of the cylinder. 5 For the cylinder in question 4, calculate the greatest possible percentage error in the surface area of the cylinder. 6 The radius of a sphere is 1.4 m with the measurement taken correct to 1 decimal place. a Calculate the volume of the sphere, correct to 1 decimal place. b Calculate the maximum possible error in the volume of the sphere. c Calculate the maximum percentage error in the volume. d Calculate the surface area of the sphere, correct to 1 decimal place. e Calculate the maximum possible error in the surface area of the sphere. f Calculate the maximum percentage error in the surface area.

72

Maths Quest General Mathematics HSC Course

WORKED

Example

18

7 An open cylindrical water tank has a radius of 45 cm and a height of 60 cm. a Calculate the capacity of the tank, in litres (correct to the nearest whole number). b If the tank’s radius is given as 50 cm, correct to the nearest 10 cm, calculate the error in the capacity of the tank. c Calculate the percentage error in the capacity of the tank. 8 A rectangular prism has dimensions 56 cm × 41 cm × 17 cm. a Calculate the volume of the prism. b Calculate the surface area of the prism. c If the dimensions are given to the nearest 10 cm, what will the dimensions of the prism be given as? d Calculate the percentage error in the volume when the dimensions are given to the nearest 10 cm. e Calculate the percentage error in the surface area when the dimensions are given to the nearest 10 cm.

Work

9 The four walls of a room are to be painted. The length of the room is 4.1 m and the width is 3.6 m. Each wall is 1.8 m high. a Calculate the area to be painted. b One litre of paint will paint an area of 2 m2. Each wall will need two coats of paint. Calculate the number of litres of paint required to complete this job. c Karla incorrectly measures the length of the room to be 3.9 m. If Karla does all her calculations using this incorrect measurement, how many litres will she be short of paint at the end of the job?

T SHEE

2.2

10 The dimensions of a rectangular house are 16.6 m by 9.8 m. a Simon takes the dimensions of the house to the nearest metre for all his calculations. What dimensions does Simon use? b Simon plans to floor the house in slate tiles. What is the area that needs to be tiled? c The tiles cost $27.50/m2 and Simon buys an extra 10% to allow for cutting and breakage. Calculate the cost of the tiles. d How much extra has Simon spent than would have been necessary had he used the original measurements of the house?

Chapter 2 Further applications of area and volume

73

summary Area of parts of the circle • The area of a circle can be calculated using the formula A = r 2. • The area of a sector is found by multiplying the area of the full circle by the fraction of the circle occupied by the sector. This is calculated by looking at the angle that the sector makes with the centre. • An annulus is the area between two circles. The area is calculated by subtracting the area of the smaller circle from the area of the larger circle or by using the formula A = (R2 – r2) , where R is the radius of the large circle and r is the radius of the small circle. • The area of an ellipse is calculated using the formula A = ab, where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Area of composite figures • The area of a composite figure is calculated by dividing the figure into two or more regular figures. • When calculating the area of a composite figure, some side lengths will need to be calculated using Pythagoras’ theorem.

Simpson’s rule • Simpson’s rule is used to find an approximation for an irregular area. • The formula for Simpson’s rule is A

h --- ( d f + 4d m + d l ) 3

.

• To obtain a better approximation for an area, Simpson’s rule can be applied twice. This is done by dividing the area in half and applying Simpson’s rule separately to each half.

Surface area of cylinders and spheres • The surface area of a closed cylinder is found by using the formula SA = 2 r 2 + 2 rh . • If the cylinder is an open cylinder, the surface area is found using SA = r 2 + 2 rh. • The surface area of a sphere is calculated using the formula SA = 4 r 2 .

Volume of composite solids • The volume of solid prisms is calculated using the formula V = A • The volume of a cone is found using the formula V =

1 --3

r 2h

• The volume of a cylinder is found using the formula V = r 2h

. .

h.

74

Maths Quest General Mathematics HSC Course

• The volume of a sphere is found using the formula V =

r3

4 --3

• The volume of a pyramid is found using the formula V = 1--- Ah 3

. .

• Other solids have their volume calculated by dividing the solid into regular solid shapes.

Error in measurement • All measurements are approximations. The maximum error in any measurement is half the smallest unit used. • Any error in a measurement will compound when further calculations using the measurement need to be made.

Chapter 2 Further applications of area and volume

75

CHAPTER review 1 Calculate the area of each of the circles below. Give each answer correct to 1 decimal place. a b c 3.7 cm

52 mm

1.7 m

2 Calculate the area of each of the figures below. Give each answer correct to 1 decimal place. a b c

92 mm

2A

2A

237° 12.5 cm

30° 4.8 m

3 Calculate the area of each of the annuluses below. Give each answer correct to 1 decimal place. a b c 34 cm

3.7 m

81 mm

17 cm

1.3 m

94 mm

4 Calculate the area of each of the ellipses below, correct to 1 decimal place. a b c 9.2 m

30 mm 45 mm

2A

2A 3.6 cm

11.4 m 7 cm

5 Calculate the area of the figure below.

15 cm

35 cm

10 cm 12 cm

10 cm

2B

76 2B

Maths Quest General Mathematics HSC Course

6 Calculate the area of each of the figures below. Where appropriate, give your answer correct to 2 decimal places. a 0.7 m b c 4.1 m

1.5 cm

7 Use Simpson’s rule to approximate the area on the right.

13 m 42 m 42 m

21 m

8 Use Simpson’s rule to find an approximation for each of the areas below. a b c

31 m

57 m

36 m 14 m

30 m

2m

62 m

96 m 24 m

42 m

50 m 30 m

30 m

9m 15 m 15 m 15 m 15 m

2D

23 m

38 m

10 Use Simpson’s rule twice to find an approximation for the area on the right.

44 m

2C

33 m

62 m

25 m

19 m 11 m

9 By dividing the area shown on the right into two sections, use Simpson’s rule to find an approximation for the area.

27 m

2C

57 m

2C

36 cm 6 cm

29 m

2C

1.5 cm

50 m

3.9 m

0.9 m

3 cm

11 Calculate the surface area of each of the closed cylinders drawn below, correct to 1 decimal place. a b c 60 cm

25 cm

10 cm

1.1 m

7 cm 4 cm

Chapter 2 Further applications of area and volume

77

12 Calculate the surface area of an open cylinder with a diameter of 9 cm and a height of 15 cm. Give your answer correct to the nearest whole number.

2D

13 Calculate the surface area of a sphere with: a a radius of 5 cm b a radius of 2.4 m Give each answer correct to the nearest whole number.

2D

c a diameter of 156 mm.

0.5 m

2E 1.9 m

3.1 m

14 Calculate the volume of the solid drawn on the right.

0.6 m 2.7 m

15 Calculate the volume of each of the solids drawn below. Where necessary, give your answer correct to the nearest whole number. a b c 12 cm

2E

22 cm 17 cm

9 cm 15 cm

20 cm

12 cm

19 cm

3 cm

10 cm

3 cm 3 cm 10 cm

40 cm

16 Calculate the volume of the figure drawn on the right, correct to 2 decimal places.

2E 15 cm 9 cm

17 A sphere has a diameter of 16 cm when measured to the nearest centimetre. a State the maximum error made in the measurement of the radius. b Calculate the volume of the sphere. Answer correct to the nearest whole number. c Calculate the maximum percentage error in the volume of the sphere.

2F

18 An aluminium soft drink can has a diameter of 8 cm and a height of 10 cm. a Calculate the capacity of the can, in millilitres, correct to the nearest 10 millilitres. b The machine that cuts the aluminium for the can is mistakenly set to 12 cm. Calculate the percentage error in the capacity of the can (correct to the nearest whole number).

2F

78

Maths Quest General Mathematics HSC Course

Practice examination questions 1 multiple choice 6.2 mm

Which of the following calculations will correctly give the area of the ellipse drawn on the right? A 6.22 B 8.52 C 10.82 D 10.8 6.2

10.8 mm

20 m

25 m

15 m

30 m

The field drawn on the right is to have its area approximated by two applications of Simpson’s rule. The value of h is: A 16 B 20 C 40 D 80

10 m

2 multiple choice

80 m

3 multiple choice The figure drawn on the right is an open cylinder. Which of the calculations below will correctly give the surface area of the cylinder? A 52 + 2 5 20 2 B 2 5 +2 5 20 C 102 + 2 10 20 D2 102 + 2 10 20

20 cm 10 cm

4 multiple choice A closed cylinder is measured as having a radius of 1.2 m and a height of 1.4 m. The maximum error in the calculation of the surface area is: A 1.2 m2 B 1.5 m2 C 1.6 m2 D 19.6 m2 5 The figure on the right shows a section of a concrete drainage pipe. a Calculate the area of the annulus, correct to 1 decimal 2.5 m place. b Calculate the volume of concrete needed to make a 1.5 m 5 m length of this pipe (correct to 1 decimal place). c Calculate the volume of water that will flow through the 5 m length of the pipe (in litres, to the nearest 100 L). d Calculate the surface area of a 5 m section of pipe (correct to the nearest m2). (Hint: Include the area of the inside of the pipe.)

CHAPTER

2

4.9 m

5.1 m

60 m 9.2 m

test yourself

6 The diagram on the right shows the cross-section of a river. a Use two applications of Simpson’s rule to find the approximate area of the river’s cross-section. b If the river flows with this cross-section for approximately 800 m, calculate the volume of the river. c The length of the river has been approximated to the nearest 100 m. Calculate the maximum percentage error in calculating this volume.

Applications of trigonometry

3 syllabus reference Measurement 6 • Applications of trigonometry

In this chapter 3A Review of right-angled triangles 3B Bearings 3C Using the sine rule to find side lengths 3D Using the sine rule to find angles 3E Area of a triangle 3F Using the cosine rule to find side lengths 3G Using the cosine rule to find angles 3H Radial surveys

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

3.1

Right-angled trigonometry — finding a side length

1 In each of the following find the length of the side marked with the pronumerals correct to two decimal places. d a b c f 40°

71°

16 m

a

19.5 m

25.2 km 63°

3.2

Using the inverse trigonometric ratios

2 Find angle , where a sin = 0.7

3.4

is acute, correct to the nearest degree. b tan = 1.5

c cos

= 0.8

Right-angled trigonometry — finding an angle

3 In each of the following find the size of the angle marked with the pronumerals correct to the nearest degree. 8m a b c 46.1 mm

20 cm

25 cm

16 m

31.2 mm

3.5

Converting nautical miles to kilometres

4 Use 1 nautical mile = 1.852 km to convert: a 4 nautical miles to kilometres. c 1.2 nautical miles to metres.

3.6

b 50 kilometres to nautical miles. d 3560 metres to nautical miles.

Angle sum of a triangle

5 Find the angle marked with the pronumeral in each of the following. a b c 132° 41° 58°

3.7

63°

71°

Solving fractional equations

6 Solve each of the following equations, where appropriate give your answer correct to 2 decimal places. x x 3 x 9.5 9 2 a --- = 3 b --- = --c ------- = ------d --- = --5 4 8 3.6 2.4 x 5

Chapter 3 Applications of trigonometry

81

Review of right-angled triangles Previously we have studied right-angled triangles and discovered that we can calculate a side length of a triangle when given the length of one other side and one of the acute angles. To do this we need to use the formulas for the three trigonometric ratios. sin cos tan

opposite side = -------------------------------hypotenuse adjacent side = --------------------------------hypotenuse opposite side = --------------------------------adjacent side

WORKED Example 1

Find the length of the side marked x in the figure on the right (correct to 1 decimal place).

42° x

29.2 cm

THINK 1

WRITE

Label the sides of the diagram. 42° adj

hyp x

29.2 cm opp

4

Choose the sine ratio and write the formula. Substitute for the opposite side and hypotenuse. Make x the subject of the formula.

5

Calculate the value of x.

2 3

opposite side = ------------------------------hypotenuse 29.2 sin 42° = ---------x x sin 42° = 29.2 29.2 x = ----------------sin 42° x = 43.6 m sin

82

Maths Quest General Mathematics HSC Course

equation solver to find Graphics Calculator tip! Using side lengths Consider worked example 1. Once you have chosen the correct trigonometric ratio and substituted, you can finish the solution using the equation solver on your graphics calculator. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any equation, enter the equation sin 42 = 29.2 ÷ X and press EXE . Note: Your calculator may display a different value of X at this stage. This is just the last value of X stored in the calculator’s memory. 4. Press F6 (SOLV) to solve the equation.

The same formulas can be used to calculate the size of an angle if we are given two side lengths in the triangle.

WORKED Example 2

Calculate the size of the angle marked in the figure on the right (correct to the nearest degree).

47 mm 35 mm

θ

THINK 1 Label the sides of the triangle.

4

Choose the tangent ratio and write the formula. Substitute for the opposite side and the adjacent side. Make the subject of the formula.

5

Calculate .

2 3

WRITE Opposite = 47 mm Adjacent = 35 mm opposite side tan = ------------------------------adjacent side 47 tan = -----35 47 = tan 1 -----35 = 53°

83

Chapter 3 Applications of trigonometry

equation solver to find Graphics Calculator tip! Using the size of an angle We can use the equation solver when we are finding the size of an angle. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any existing equation, then enter the equation tan X = 47 ab/c 35 and press EXE . Note: Your calculator may display a different value of X at this stage. This is just the last value of X stored in the calculator’s memory. 4. Press F6 (SOLV) to solve the equation.

Using these results, we are able to solve problems that involve more than one rightangled triangle.

WORKED Example 3 Greg stands 70 m from the base of a building and measures the angle of elevation to the top of the building as being 35°. Julie is standing 40 m from the base of the building on the other side of the building as shown in the figure on the right. a Calculate the height of the building, correct to 2 decimal places. b Calculate the angle of elevation of the top of the building that Julie would measure, correct to the nearest degree. THINK a

1

Draw the triangle showing the angle of elevation from where Greg is standing and label the sides.

h 35° 40 m

70 m

WRITE a h

35° 70 m Continued over page

84

Maths Quest General Mathematics HSC Course

b

WRITE

2

Choose the tangent ratio and write the formula.

3

4

Substitute for and the adjacent side. Make h the subject of the formula.

5

Calculate the value of h.

1

Draw the triangle from where Julie is standing and label the sides.

tan

opposite side = ------------------------------adjacent side

h tan 35° = -----70 h = 70

tan 35°

h = 49.01 m b 49.01 m

THINK

θ 40 m 2

Choose the tangent ratio and write the formula.

tan

opposite side = ------------------------------adjacent side

3

Substitute for the opposite side and the adjacent side.

tan

49.01 = ------------40

4

Make

5

Calculate , correct to the nearest degree.

the subject of the formula.

= tan

1

49.01 ------------40

= 51°

remember 1. The formulas for the three trigonometric ratios are: opposite side • sin = ------------------------------hypotenuse adjacent side • cos = ------------------------------hypotenuse opposite side • tan = ------------------------------adjacent side 2. To calculate the length of a side we need to be given one side length and one acute angle. 3. To calculate the size of an angle we need to be given two side lengths. 4. Many problems involve solving two or more right-angled triangles. 5. After substitution, the value of the unknown can be found using the equation solver on a graphics calculator.

85

Chapter 3 Applications of trigonometry

3A

Review of right-angled triangles

1 Calculate the length of the side marked with the pronumerals in each of the following, 3.1 correct to 1 decimal place. 1 a b c Right-angled b

WORKED

Example

38°

142 mm

23° a

trigonometry — finding a side length

c

61°

Cabri Geo

e

314 mm

ry met

11.4 m

d

HEET

13.2 cm

SkillS

Sine, cosine and tangent

f 17° 5° e

d 50°

f

19.2 cm

9.1 m

2 Calculate the size of each of the angles marked with the pronumerals, correct to the nearest degree. 2 a b c 113 cm 9.5 m

WORKED

cm

θ 61

θ 71 mm

11.4 m

3.2

SkillS HEET

36 mm

Example

Using the inverse trigonometric ratios

θ

3 From the top of a cliff the angle of depression to a boat sailing 100 m offshore is 32°. Calculate the height of the cliff, correct to the nearest metre.

100 m 32° h

SkillS

Rounding angles to the nearest degree

3.4

SkillS HEET

5 A lighthouse is 40 m tall and the beacon atop the lighthouse is sighted by a ship 150 m from shore, as shown in the figure on the right. Calculate the angle 40 m of elevation at which the lighthouse is sighted, correct to the nearest degree.

3.3

HEET

4 Andrew walks 5 km from point P to point Q. At the same time Bianca walks from P to R such that PQ is perpendicular to PR. Given that –PQR = 28°: a draw a diagram of DPQR b calculate the distance walked by Bianca, correct to the nearest metre c calculate the distance that Andrew would need to walk in a straight line to meet Bianca, correct to the nearest metre.

θ 150 m

Right-angled trigonometry — finding an angle

86

Maths Quest General Mathematics HSC Course

6 From a point 65 m above the ground, a point is sighted on the ground at a distance of 239 m. a Draw a diagram of this situation. b Calculate the angle of depression at which the point is sighted. 7 Sally and Tim are both sighting the top of a building, as shown in the figure on the right. 3 Sally is 40 m from the base of the building and sights the angle of elevation to the top of the building as 35°. Tim is 60 m from the base of the building. a Calculate the height of the building, correct to 2 decimal places. b Calculate the angle of elevation at which Tim will sight the building.

WORKED

Example

8 George and Diego are both flying a kite from the same point. George’s kite is flying on 50 m of string and the string makes a 70° angle with the ground. Diego’s kite is flying on a 60 m piece of string and is at the same height as George’s kite, as shown in the figure on the right. Calculate the angle that the string from Diego’s kite makes with the ground. Give your answer correct to the nearest degree.

h

35° 40 m

60 m

S

D

G

50 m

60 m

70°

Bearings A bearing is an angle used to describe direction. Bearings are used in navigation and are a common application of trigonometry to practical situations. We can therefore apply our trigonometrical formulas to make calculations based upon these bearings. There are two types of bearing that we need to be able to work with: compass bearings and true bearings.

Compass bearings Compass bearings use the four points of the compass. With compass bearings there are four main directions: north, south, east and west. In between each of these main directions there are four others: north-east, south-east, south-west and north-west. Each of these directions is at 45° to two of the four main directions. Trigonometry can then be used to solve problems about distances and angles using these eight basic directions.

N NW

NE

W

E

SW

SE S

Chapter 3 Applications of trigonometry

87

WORKED Example 4

A ship (A) is 10 nautical miles due east of a lighthouse. A second ship (B) bears SE of the lighthouse and is due south of the first ship. Calculate the distance of the second ship from the lighthouse, correct to 1 decimal place. THINK 1

WRITE

Draw a diagram labelling the sides of the triangle.

adj 10 M

N L

A

45°

opp

hyp x B 2

3

4

5

6

Choose the cosine ratio and write the formula. Substitute for and the adjacent side. Make x the subject of the equation.

Calculate the value of x, correct to 1 decimal place. Give a written answer.

adj = --------hyp 10 cos 45° = -----x x cos 45° = 10 10 x = -----------------cos 45° = 14.1 M cos

The second ship is 14.1 nautical miles from the lighthouse.

These eight compass points do not allow us to make calculations about more precise directions. For this reason an alternative method of describing bearings is needed for any direction other than these basic eight points.

True bearings A true bearing is an angle measured from north in a clockwise direction. As there are 360° in a revolution, all true bearings are represented as a three-digit number between 000° and 360°. For example, east is at a bearing of 090°, south has a bearing of 180° and west 270°. When given information about a bearing, we can solve problems using trigonometry by constructing a right-angled triangle. As most questions involving bearings are in problem form, a diagram is necessary to solve the problem and an answer in words should be given.

N

270°

090°

180°

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Maths Quest General Mathematics HSC Course

WORKED Example 5 A ship sails on a bearing of 130° for a distance of 10 nautical miles. Calculate how far south of its starting point the ship is, correct to 2 decimal places. THINK 1

WRITE

Draw a diagram completing a right-angled triangle and label the sides.

N 130° 50°

hyp 10 M

adj x

opp

3

Choose the cosine ratio and write the formula. Substitute for and the hypotenuse.

4

Make x the subject of the equation.

5

Calculate. Give a written answer.

2

6

adj = --------hyp x cos 50° = -----10 x = 10 cos 50° x = 6.43 M The ship is 6.43 nautical miles south of its starting point. cos

We can also use our methods of calculating angles to make calculations about bearings. After solving the right-angled triangle, however, we need to provide the answer as a bearing.

WORKED Example 6 On a hike Lisa walked south for 3.5 km and then turned west for 1.2 km. Calculate Lisa’s bearing from her starting point.

1

WRITE

Draw a diagram and label the sides of the triangle.

N

θ hyp

3.5 km adj

THINK

1.2 km opp 2

Choose the tangent ratio and write the formula.

tan

opp = --------adj

Chapter 3 Applications of trigonometry

THINK

89

WRITE

3

Substitute for the opposite and adjacent sides and simplify.

4

Make

5

Calculate .

6

From the diagram we can see the angle lies between south and west. South has a bearing of 180°, and so we must add 19° to 180° to calculate the true bearing.

Bearing = 180° + 19° = 199°

7

Give a written answer.

Lisa is at a bearing of 199° from her starting point.

tan

1.2 = ------3.5 = 0.3429 = tan 1(0.3429)

the subject of the equation.

= 19°

remember 1. Bearings are used to describe a direction. We have used two types of bearings. • Compass bearings use the four main points of the compass, north, south, east and west, as well as the four middle directions, north-east, north-west, southeast and south-west. • True bearings describe more specific direction by using a three-digit angle, which is measured from north in a clockwise direction. 2. Bearing questions are usually given in written form so you will need to draw a diagram to extract all the information from the question. 3. Read carefully to see if the question is asking you to find a side or an angle. 4. Always give a written answer to worded questions. 5. Use 1 M = 1.852 km to convert beteen nautical miles and kilometres.

3B WORKED

Example

1 A road runs due north. A hiker leaves the road and walks for 4.2 km in a NW direction. a Draw a diagram of this situation. b How far due east must the hiker walk to get back to the road? (Give your answer correct to 3 decimal places.) 2 A driver heads due south for 34 km, then turns left and drives until he is SE of his starting point. a Draw a diagram to show the driver’s journey. b Calculate the distance the driver travelled in an easterly direction from his starting point.

3.5

SkillS

HEET

4

Bearings

Converting nautical miles to kilometres

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Maths Quest General Mathematics HSC Course

3 Two boats, A and B, sail from a port. A heads due west, while B heads NW for a distance of 43 nautical miles, where it drops anchor. Boat A drops anchor due south of boat B. a Draw a diagram showing the positions of boats A and B. b Calculate the distance between boats A and B in nautical miles, correct to 1 decimal place. c Calculate the distance in kilometres between A and B. 4 multiple choice A true bearing of 315° is equivalent to a compass bearing of: A NE B NW C SE D SW 5 multiple choice A compass bearing of SE is equivalent to a true bearing of: A 045° B 135° C 225° Example

5

6 Two hikers, Adrian and Bertrand, set out on a walk. Adrian walks 5 km due north to a point, A, and Bertrand walks on a bearing of 052° to a point, B. Bertrand lets off a flare and Adrian notices Bertrand is now due east of him, as shown in the diagram on the right. Calculate the distance between the two hikers, correct to 1 decimal place.

A

d

B

5 km

WORKED

D 315°

52°

7 A yacht sights a lighthouse on a bearing of 060°. After sailing another eight nautical miles due north, the yacht is due west of the lighthouse. a Draw a diagram of this situation. b Calculate the distance from the yacht to the lighthouse when it is due west of it (correct to 1 decimal place). 8 An aeroplane takes off from an airport and flies on a bearing of 220° for a distance of 570 km. Calculate how far south of the airport the aeroplane is (correct to the nearest kilometre). 9 A camping ground is due east of a car park. Eden and Jeff walk 3.8 km due south from the camping ground until the car park is on a bearing of 290°. a Draw a diagram showing the car park, the camping ground, and Eden and Jeff’s position. b Calculate the distance Eden and Jeff need to walk directly back to the car park, correct to 1 decimal place. 10 multiple choice A ship is on a bearing of 070° from a lighthouse. The bearing of the lighthouse from the ship will be: A 070° B 160° C 200° D 250°

91

Chapter 3 Applications of trigonometry

11 multiple choice A camping ground is SW of a car park. The bearing of the car park from the camping ground will be: A NE B NW C SE D SW 12 A search party leaves its base and head 4 km due west before turning south for 3.5 km. a Draw a diagram of this situation. 6 b Calculate the true bearing of the search party from its base, correct to the nearest degree.

WORKED

Example

13 A ship is two nautical miles due west of a harbour. A yacht that sails 6.5 nautical miles from that harbour is due north of the ship. Calculate the true bearing (correct to the nearest degree) of the course on which the yacht sails from the harbour.

Trigonometric ratios for obtuse angles Many non-right-angled triangles have one obtuse angle. In the following sections we will be solving non-right-angled triangles and will need to investigate the trigonometric ratios for obtuse angles. 1 Use your calculator to give each of the following, correct to 3 decimal places. a sin 100° b cos 100° c tan 100° d sin 135° e cos 135° f tan 135° g sin 179° h cos 179° i tan 179° 2 Which of the answers to question 1 are positive and which are negative? 3 Calculate the sine, cosine and tangent of several other obtuse angles and see if the established pattern continues. 4 Can you develop a rule for the sign of trigonometric ratios of obtuse angles?

The sine rule Finding side lengths The trigonometry we have studied so far has been applicable to only right-angled triangles. The sine rule allows us to calculate the lengths of sides and the size of angles in non-right-angled triangles. Consider the triangle drawn on the right.

C C b

a

A A

B c

B

The sine rule states that in any triangle, ABC, the ratio of each side to the sine of its opposite angle will be equal. a b c ------------- = ------------- = ------------sin A sin B sin C

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Maths Quest General Mathematics HSC Course

Derivation of the sine rule A, B and C represent the three angles in the triangle ABC and a, b and c represent the three sides, remembering that each side is named with the lower-case letter of the opposite vertex. Construct a line from C to a point, D, perpendicular to AB. CD is the perpendicular height of the triangle, h. C

b

a h

A

B

A

B

D c

Now consider ACD and BCD separately. C

C

b

A

a h

h

D

D

B

Use the formula for the sine ratio: opp opp = --------sin = --------hyp hyp h h sin A = --sin B = --b a h = b sin A h = a sin B We are now able to equate these two expressions for h. a sin B = b sin A Dividing both sides by sin A sin B we get: sin

a sin B b sin A --------------------------- = --------------------------sin A sin B sin A sin B a b ------------ = -----------sin A sin B c Similarly, we are able to show that each of these is also equal to ------------- . Try it! sin C

This formula allows us to calculate the length of a side in any triangle if we are given the length of one other side and two angles. When using the formula we need to use only two parts of it.

Chapter 3 Applications of trigonometry

93

WORKED Example 7

Calculate the length of the side marked x in the triangle on the right, correct to 1 decimal place.

A 80° 16 cm 40° B

THINK

C

WRITE

1

Write the formula.

2

Substitute a = x, b = 16, A = 80° and B = 40°.

3

Make x the subject of the equation by multiplying by sin 80°. Calculate.

4

x

a b ------------ = -----------sin A sin B x 16 ----------------- = ----------------sin 80° sin 40° 16 sin 80° x = ------------------------sin 40° x = 24.5 cm

equation solver to solve Graphics Calculator tip! Using sine rule problems (sides) As with right-angled trigonometry, you can use the equation solver function on your graphics calculator to solve the equation formed immediately after you substitute into the equation. Consider worked example 7 above. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any existing equation, enter the equation X ÷ sin 80 = 16 ÷ sin 40, and then press EXE . Note: Your calculator may display a different value of X at this stage. This is just the last value of X stored in the calculator’s memory. 4. Press F6 (SOLV) to solve the equation.

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Maths Quest General Mathematics HSC Course

Note: Some questions may ask for you to give the answer in a form other than a number and as such the graphics calculator method can not be used. For example, the 16 sin 80° question above could be worded to, say, show x = ------------------------ , in which case you must sin 40° manipulate the equation to arrive at the desired expression. To use the sine rule we need to know the angle opposite the side we are finding and the angle opposite the side we are given. In some cases these are not the angles we are given. In such cases we need to use the fact that the angles in a triangle add to 180° to calculate the required angle.

WORKED Example 8

A

Calculate the length of the side labelled m in the figure on the right, correct to 4 significant figures.

65° m

75° B

THINK 1 Calculate the size of angle C. 2

Write the formula.

3 4

Substitute a = 16, c = m, A = 65° and C = 40°. Make m the subject of the equation.

5

Calculate.

16 m

C

WRITE C = 180° 65° = 40° a c ------------ = ------------sin A sin C 16 m ----------------- = ----------------sin 65° sin 40° 16 sin 40° m = ------------------------sin 65° = 11.35 m

75°

As mentioned in the previous investigation, we need to apply the sine rule to obtuseangled triangles. In such examples the method used is exactly the same with the substitution of an obtuse angle. Using the sine rule allows us to solve a number of more complex problems. As with our earlier trigonometry problems, we begin each with a diagram and give a written answer to each.

WORKED Example 9

A

Georg looks south and observes an aeroplane at an angle of elevation of 60°. Henrietta is 20 km south of where Georg is and she faces north to see the aeroplane at an angle of elevation of 75°. Calculate the distance of the aeroplane from Henrietta’s observation point, to the nearest metre. 60° G

THINK 1 Calculate the size of

GAH.

WRITE A = 180° = 45°

x

75° 20 km

60°

75°

H

Chapter 3 Applications of trigonometry

THINK

95

WRITE

2

Write the formula.

3

Substitute g = x, a = 20, G = 60° and H = 75°.

4 5

Make x the subject. Calculate.

6

Give a written answer.

g a ------------- = -----------sin G sin A x 20 ----------------- = ----------------sin 60° sin 45° 20 sin 60° x = ------------------------sin 45° x = 24.495 km The distance of the aeroplane from Henrietta’s observation point is 24.495 km.

remember a b c 1. The sine rule formula is ------------ = ------------ = ------------- . sin A sin B sin C 2. The sine rule is used to find a side in any triangle when we are given the length of one other side and two angles. 3. We need to use only two parts of the sine rule formula. 4. For written problems, begin by drawing a diagram and finish by giving a written answer. 5. You can use the equation solver on a graphics calculator to find the value of the unknown after substituting into the formula.

Using the sine rule to find side lengths

3C

1 Write down the sine rule formula as it applies to each of the triangles below. a b X c P A

WORKED

Example

C

a

Z

Y

R

Q

2 Use the sine rule to calculate the length of the side marked with the pronumeral in each of the following, correct to 3 significant figures. a b c L A R x 50° B

16 cm

1.9 km

63°

t

45° C M

52°

q

59°

84° N

T

89 mm

S

3.7

SkillS

HEET

7

Angle sum of a triangle

b

B

SkillS

HEET

c

3.6

Solving fractional equations

96 Triangle

WORKED

Example

8

3 In each of the following, use the sine rule to calculate the length of the side marked with the pronumeral, correct to 1 decimal place, by first finding the size of the third angle. a G b c x H B N 74° 74° 80° 18.2 mm

19.4 km

Cabr

omet i Ge ry

Maths Quest General Mathematics HSC Course

m 62° P

85°

y 27°

C

A 35.3 cm I

M

4 multiple choice 42 cm Look at the figure drawn on the right. Which of the following expressions gives 28° 35° m the value of m? 42 sin 117° 42 sin 117° A m = ---------------------------B m = ---------------------------sin 28° sin 35°

42 sin 28° C m = ------------------------sin 117°

42 sin 35° D m = ------------------------sin 117°

5 multiple choice Look at the figure drawn on the right. Which of the following expressions gives the value of n?

n 28°

35° 42 m

42 sin 117° A n = ---------------------------sin 28°

42 sin 117° B n = ---------------------------sin 35°

42 sin 28° C n = ------------------------sin 117°

42 sin 35° D n = ------------------------sin 117°

6 ABC is a triangle in which BC = 9 cm, –BAC = 54° and –ACB = 62°. Calculate the length of side AB, correct to 1 decimal place. 7 XYZ is a triangle in which y = 19.2 m, –XYZ = 42° and –XZY = 28°. Calculate x, correct to 3 significant figures. WORKED

Example

9

8 X and Y are two trees, 30 m apart on one side of a river. Z is a tree on the opposite side of the river, as shown in the diagram below. Z

59° X

72° 30 m

Y

It is found that –XYZ = 72° and –YXZ = 59°. Calculate the distance XZ, correct to 1 decimal place.

97

Chapter 3 Applications of trigonometry

9 From a point, M, the angle of elevation to the top of a building, B, is 34°. From a point, N, 20 m closer to the building, the angle of elevation is 49°. a Draw a diagram of this situation. b Calculate the distance NB, correct to 1 decimal place. c Calculate the height of the building, correct to the nearest metre. 10 Look at the figure on the right. a Show that XY can be given by the 80 sin 30° expression ------------------------- . sin 40° b Show that h can be found using the 80 sin 30° sin 70° expression -------------------------------------------- . sin 40° c Calculate h, correct to 1 decimal place.

Y

h 30° 80 m

W

70° X

Z

Finding angles Using the sine rule result, we are able to calculate angle sizes as well. To do this, we need to be given the length of two sides and the angle opposite one of them. For simplicity, in solving the triangle we invert the sine rule formula when we are using it to find an angle. The formula is written: sin A sin B sin C ------------ = ------------ = ------------a b c Your formula sheet has the sine rule to find a side length. You need to invert this formula when finding an angle. As with finding side lengths, we use only two parts of the formula.

WORKED Example 10 Find the size of the angle, , in the figure on the right, correct to the nearest degree.

A 6 cm

110° C 20 cm

B

THINK 1

Write the formula.

2

Substitute A = 110°, C = , a = 20 and c = 6.

3 4

Make sin the subject of the equation. Calculate a value for sin .

5

Calculate sin 1(0.2819) to find .

WRITE sin A sin C ------------ = ------------a c sin 110° sin -------------------- = -----------20 6 6 sin 110° sin = ------------------------20 sin = 0.2819 = 16°

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Maths Quest General Mathematics HSC Course

equation solver to solve Graphics Calculator tip! Using sine rule problems (angles) The same graphics calculator method can be used when finding an angle using the sine rule. Consider worked example 10. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any existing equation, enter the equation sin 110˚ ÷ 20 = sin X ÷ 6 and press EXE . Note: Your calculator may display a different value of X at this stage. This is just the last value of X stored in the calculators memory. 4. Press F6 (SOLV) to solve the equation.

Note: When using the graphics calculator, you do not need to remember to invert the sine rule. If you enter 20 ÷ sin 110 = 6 sin x, the graphics calculator will still solve the equation. As with finding side lengths, some questions will be problems that require you to draw a diagram to extract the required information and then write the answer.

WORKED Example 11

From a point, P, a ship (S) is sighted 12.4 km from P on a bearing of 137°. A point, Q, is due south of P and is a distance of 31.2 km from the ship. Calculate the bearing of the ship from Q, correct to the nearest degree. THINK 1

Draw a diagram.

WRITE P

137° 12.4 km 43° S

31.2 km

Q

Chapter 3 Applications of trigonometry

THINK

99

WRITE

2

Write the formula.

3

Substitute for p, q and P.

4

Make sin Q the subject.

5

Calculate a value for sin Q.

sin Q sin P ------------- = -----------q p sin Q sin 43° ------------- = ----------------12.4 31.2 12.4 sin 43° sin Q = -----------------------------31.2 sin Q = 0.271

1

6

Calculate sin (0.271) to find Q.

7

Give a written answer.

Q = 16° The bearing of the ship from Q is 016°.

remember sin A sin B sin C 1. The sine rule formula for finding an angle is ------------ = ------------ = ------------- . a b c 2. The formula sheet gives the sine rule in the form used to find a side. You have to invert the formula when finding angles. 3. We can use this formula when we are given two sides and the angle opposite one of them. 4. Worded questions should begin with a diagram and finish with a written answer.

Using the sine rule to find angles

3D

1 Find the size of the angle marked with a pronumeral in each of the following, correct to the nearest degree. 10 a b c L P A

WORKED

Example

32 cm

100°

29.5 m B

46 cm

d

153 mm 79 mm

C R Q 60° 18.9 m

e

V

M

117°

f

X

N 27 mm

23.6 km

U

75°

23.6 km

W

16.5 cm

170°

Y

27.6 cm 86°

Z

156 mm

100

Maths Quest General Mathematics HSC Course

2 multiple choice

36°

36 sin 13° = ------------------------7

C sin

13

7

Which of the statements below give the correct value for sin ? 13 sin 36° A sin = ------------------------7

θ

B sin

7 sin 36° = ---------------------13

D sin

7 sin 13° = ---------------------36

3 multiple choice In which of the triangles below is the information insufficient to use the sine rule? A

B

θ

θ

12.7 m

14.8 m

45°

57° 16.2 m

12.6 m

C

D 115°

12.7 m

6.2 m

8.7 m

θ

9°

θ 12.9 m

4 In PQR, q = 12 cm, r = 16 cm and the nearest degree.

PRQ = 56°. Find the size of

5 In KLM, LM = 4.2 m, KL = 5.6 m and to the nearest degree.

KML = 27°. Find the size of

PQR, correct to

LKM, correct

6 A, B and C are three towns marked on a map. Judy calculates that the distance between A and B is 45 km and the distance between B and C is 32 km. CAB is 45°. Calculate 11 ACB, correct to the nearest degree.

WORKED

Example

7 A surveyor marks three points X, Y and Z in the ground. The surveyor measures XY to be 13.7 m and XZ to be 14.2 m. XYZ is 60°. a Calculate XZY to the nearest degree. b Calculate YXZ to the nearest degree.

Work

8 Two wires support a flagpole. The first wire is 8 m long and makes a 65° angle with the ground. The second wire is 9 m long. Find the angle that the second wire makes with the ground. T SHEE

3.1

101

Chapter 3 Applications of trigonometry

1 1 Find a in the triangle below, correct to 1 decimal place.

2 Find b in the triangle below, correct to the nearest millimetre.

23° a

346 mm 11.4 m

63° b

3 Find c in the triangle below, correct to 3 significant figures.

4 Find θ in the triangle below, correct to the nearest degree.

42 cm 12 m 37° c 7m

θ

In questions 5 to 7 find the size of the side marked with a pronumeral, correct to 2 significant figures. 5

6 80°

7

46 m

12°

y

x

z

75°

150° 6.1 cm

23°

30° 1700 mm

In questions 8 to 10 find the size of the angle marked θ, correct to the nearest degree. 8

9

10 44 cm

θ

65 cm 41 m

θ

23° 4.9 m

31° 60° 32 m

θ 3.6 m

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Maths Quest General Mathematics HSC Course

Area of a triangle You should be familiar with finding the area of a triangle using the formula Area = 1--2- bh . In this formula, b is the base of the triangle and h is the perpendicular height. This formula can’t be used in triangles where we do not know the perpendicular height. Trigonometry allows us to find the area of such triangles when we are given the length of two sides and the B included angle. Consider the triangle drawn on the right. In this triangle: Area = 1--2- ah

A

c

b h C

D a

[1]

(a = base of triangle, h = height) Now consider ACD. Since this triangle is right angled: opp sin C = --------hyp h sin C = --b h = b sin C Substituting for h in [1]: Area = 1--2- ab sin C This becomes the formula for the area of a triangle. There are three equivalent formulas for the area of a triangle. Area = 1--2- ab sin C Area = 1--2- ac sin B Area = 1--2- bc sin A The formula sheet gives the first version of this formula only. The others are an adaptation of the same rule. These formulas allow us to find the area of any triangle where we are given the length of two sides and the included angle. The included angle is the angle between the two given sides. The formula chosen should be the one that uses the angle you have been given.

WORKED Example 12

A

Find the area of the triangle on the right, correct to 2 decimal places.

12 cm

60°

B

THINK 2

Write the formula that uses sin B. Substitute a = 16, c = 12 and B = 60°.

3

Calculate.

1

16 cm

WRITE Area = 1--- ac sin B Area =

2 1 --2

16

Area = 83.14 cm

12 2

sin 60°

C

Chapter 3 Applications of trigonometry

103

As with all other trigonometry we can use this formula to solve practical problems.

WORKED Example 13 Two paths diverge at an angle of 72°. The paths’ lengths are 45 m and 76 m respectively. Calculate the area between the two paths, correct to the nearest square metre. THINK 1

WRITE

Draw a diagram. 45 m 76 m

72° 2

Write the formula.

Area = 1--- ab sin C

3

Substitute a = 45, b = 76 and C = 72°.

Area =

4

Calculate.

5

Give a written answer.

Area = 1626 m2 The area between the paths is 1626 m2.

2 1 --2

¥ 45 ¥ 76 ¥ sin 72°

remember 1. The area of a triangle can be found when you are given the length of two sides and an included angle. 2. The formulas to use are:

Area = 1--- ab sin C 2

Area = 1--- ac sin B 2

Area = 1--- bc sin A 2

3. Where possible you should still use Area = 1--- bh. 2

4. Begin worded problems with a diagram and finish them with a written answer.

3E

Area of a triangle

1 Write down the formula for the area of a triangle in terms of each of the triangles drawn below. Write the formula using the boldfaced angle. a b X c B A

A

C

Y

Z

G

M

104

Maths Quest General Mathematics HSC Course

2 For each of the triangles drawn below, state whether the area would be best found using the formula Area = 1--- ab sin C or Area = 1--- bh. 2

2

a

b 6 cm 1.9 m

60° 12 cm

2.6 m

c

d 6.2 m 9.1 m

8.3 m

60° 12.4 m Example

12

3 Find the area of each of the following triangles, correct to 1 decimal place. a b c 11 cm 196 mm

207 mm 117 mm

40°

120° 92 mm

10°

12 cm

4 Use either Area = 1--- ab sin C or Area = 1--- bh to find the area of each of the following 2 2 triangles. Where necessary, give your answer correct to 1 decimal place. a b c 32 cm

38 cm

WORKED

19 cm 66° 14 cm

38 cm

32 cm

5 multiple choice In which of the following triangles can the formula Area = 1--- ab sin C not be used to 2 find the area of the triangle? A B 4 cm 4 cm 60° 9 cm 9 cm

C

D 4 cm

75°

9 cm 120° 4 cm

9 cm

Chapter 3 Applications of trigonometry

105

6 multiple choice The area of the triangle on the right (correct to 1 decimal place) is: A 4.4 cm2 B 14.7 cm2 C 17.1 cm2 D 20.5 cm2

5 cm

7 In PQR, p = 4.3 cm, q = 1.8 cm and correct to 4 significant figures.

7 cm

78° 6 cm

PRQ = 87°. Calculate the area of PQR,

8 The figure on the right is of a parallelogram, ABCD. a Copy the diagram into your workbook and draw the diagonal AC on your diagram. b By considering the parallelogram as two equal triangles, calculate its area, correct to 1 decimal place.

A

B

2.5 m 70° D

5.2 m

9 On the right is a diagram of a pentagon inscribed in a circle of radius 5 cm. a Calculate the size of each of the angles made at the centre. b Calculate the area of the pentagon, correct to the nearest square centimetre. 10 A surveyor sights the four corners of a block of land and makes the following 13 notebook entry. Calculate the area of the block of land, correct to the nearest square metre.

WORKED

Example

18 m 20 m 90° 80° 70° 15 m 120° 25 m

C

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Maths Quest General Mathematics HSC Course

The cosine rule Finding side lengths When given the length of one side and two angles in a triangle, we can use the sine rule to find another side length. However, in many cases we do not have this information and need another method of calculating the side lengths. The cosine rule allows us to calculate the length of the third side of a triangle when we are given the length of the other two sides and the included angle. a2 = b2 + c2

2bc cos A

2

2

2

2ac cos B

2

2

2

2ab cos C

b =a +c c =a +b

The formula sheet gives the third version of this formula only. The others are an adaptation of the same rule. It is important to notice that the formula is given in terms of a2, b2 or c2. This means that to find the value of a, b or c we need to take the square root of our calculation.

Derivation of the cosine rule Consider ABC on the right. In this triangle, h is the perpendicular height of the triangle and meets AB at D. We will let AD = x, and therefore BD = c x.

Using Pythagoras’ theorem on BCD: From ACD: Therefore:

Therefore:

a

b h

B

c– x

a2 = (c a2 = c2

x

D c

A

x)2 + h2 2cx + x2 + h2 [1]

b2 = x2 + h2 h2 = b2 x2

Substituting for h2 in [1]:

Now in ACD:

C

a2 = c2 a2 = c2

2cx + x2 + b2 x2 2cx + b2 [2]

x cos A = --b x = b cos A

Substituting for x in [2]:

a2 = c2 2c(b cos A) + b2 a2 = c2 + b2 2bc cos A

This becomes the formula for the cosine rule. A similar formula can be used for finding sides b and c. You may like to try it for yourself. 1 Start with ABC and draw a perpendicular line from A to BC. 2 Use this diagram and follow the method shown to obtain the following version of the cosine rule: b2 = a 2 + c 2 2ac cosB. 3 Can you obtain c2 = a2 + b 2

2ab cosC?

Chapter 3 Applications of trigonometry

WORKED Example 14

B

Find the length of the side marked b in the triangle on the right, correct to 1 decimal place.

70° 10 m

12 m

A

THINK 1

2 3 4

107

C

b

WRITE

Write the formula with b2 as the subject. Substitute a = 12, c = 10 and B = 70°. Calculate the value of b2. Find b by taking the square root of b2.

b2 = a2 + c2

2ac cos B

= 122 + 102 2 = 161.915 b = 161.915 = 12.7 m

12

10

cos 70°

equation solver to solve Graphics Calculator tip! Using cosine rule problems (sides) Using the equation solver method for the cosine rule is a very useful method as many students forget the final step of the solution, which is to take the square root of a2, b2 or c2. In the same way as with earlier questions we write the formula and then substitute the appropriate values, leaving one unknown. Hence we have an equation, which can be typed into the equation solver of the graphics calculator. Consider worked example 14 above. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any existing equation, enter B2 = 122 + 102 – 2 12 10 cos 70 and then press EXE .

4. Press F6 (SOLV) to solve the equation.

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As with sine rule questions, we can apply the cosine rule to obtuse-angled triangles. You should recall from the earlier investigation that the cosine ratio of an obtuse angle is negative. The method of solution remains unchanged.

WORKED Example 15 Find the length of side PQ in the triangle on the right, correct to the nearest millimetre.

P

68 mm 122° R

THINK 1

2 3 4

Q

92 mm

WRITE

Write the formula with r 2 as the subject. Substitute p = 92, q = 68 and R = 122°. Calculate the value of r 2. Find r by taking the square root of r 2.

r 2 = p2 + q2

2pq cos R

= 922 + 682 2 = 19 718.35 r = 19 718.35 = 140 mm

92

68

cos 122°

The cosine rule also allows us to solve a wider range of practical problems. The important part of solving such problems is marking the correct information on your diagram. If you can identify two side lengths and the included angle, you can use the cosine rule.

WORKED Example 16 A surveyor standing at a point, X, sights a point, M, 50 m away and a point, N, 80 m away. If the angle between the lines XM and XN is 45°, calculate the distance between the points M and N, correct to 1 decimal place. THINK 1

WRITE

Draw a diagram and mark all given information on it.

X 45° 50 m

80 m N

M 2

3 4 5

6

Write the formula with x2 as the subject. Substitute m = 80, n = 50 and X = 45°. Calculate the value of x2. Calculate x by taking the square root of x2. Give a written answer.

x2 = m2 + n2

2mn cos X

= 802 + 502 2 = 3243.15 x = 3243.15 = 56.9 m

80

50

cos 45°

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Chapter 3 Applications of trigonometry

remember 1. To use the cosine rule to find a side length, you need to be given the length of two sides and the included angle. 2. The cosine rule formulas are: • a2 = b2 + c2 - 2bc cos A • b2 = a2 + c2 - 2ac cos B • c2 = a2 + b2 - 2ab cos C. 3. In the solution to cosine rule questions, your final answer is found by taking the square root of the calculation. 4. Begin worded questions by drawing a diagram and finish them by giving a written answer.

Using the cosine rule to find side lengths

3F

1 Write down the cosine rule formula as it applies to each of the triangles below. In each case, make the boldfaced pronumeral the subject. a

b

A

c

P q

r c

b

n

L

M

m l

B

Q

C

a

R

p

N WORKED

Example

14

2 Find the length of the side marked with a pronumeral in each of the following, correct to 3 significant figures. a

b

A x

12 m

12 m 42°

Q

14 m

B

X

13 cm

r C

35°

c

P

21 cm

Example

15

12 m

R Y

WORKED

60°

x

Z

3 In each of the following obtuse-angled triangles, find the length of the side marked with the pronumeral, correct to 1 decimal place. a

bA

X 112 cm

110° x

6.1 m B

130° 9.7 m

R q

b Z

Y

c

114 cm

C P

160° 43 mm Q

63 mm

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Maths Quest General Mathematics HSC Course

4 multiple choice In which of the following triangles are we unable to use the cosine rule to find x? A B 14.8 cm 16.4 m

132°

16.2 cm

x

x 32° 18.2 m

C

D 63° 8.3 km

9.6 km

10.5 m

9.7 m

63° x

x

5 multiple choice Look at the triangle drawn on the right. The value of x, correct to 1 decimal place, is: A 7.2 m B 7.3 m C 52.4 m D 52.5 m 6 multiple choice

50° 8m

9m

x

Lieng is asked to find the value of a, correct to 1 decimal place, in the figure drawn on the right. Below is Lieng’s solution. Line 1: a2 = 122 + 82 2 12 8 cos 60° 60° Line 2: = 144 + 64 192 cos 60° 8 cm Line 3: = 208 192 cos 60° Line 4: = 16 cos 60° Line 5: =8 a Line 6: a = 2.8 m Lieng’s solution is incorrect. In which line did she make her error? A Line 2 B Line 3 C Line 4 D Line 5

12 cm

7 In ABC, a = 14 cm, c = 25 cm and ABC = 29°. Calculate b, correct to 1 decimal place. 8 In PQR, PQ = 234 mm, QR = 981 mm and PR, correct to 3 significant figures. WORKED

Example

16

PQR = 128°. Find the length of side

9 Len and Morag walk separate paths that diverge from one another at an angle of 48°. After three hours Len has walked 7.9 km and Morag 8.6 km. Find the distance between the two walkers at this time, correct to the nearest metre. 10 A cricketer is fielding 20 m from the batsman and at an angle of 35° to the pitch. The batsman hits a ball 55 m and straight behind the bowler. How far must the fieldsman run to field the ball? (Give your answer to the nearest metre.) 11 The sides of a parallelogram are 5.3 cm and 11.3 cm. The sides meet at angles of 134° and 46°. a Draw a diagram of the parallelogram showing this information and mark both diagonals on it. b Calculate the length of the shorter diagonal, correct to 1 decimal place. c Calculate the length of the long diagonal, correct to 1 decimal place.

Chapter 3 Applications of trigonometry

12 The cord supporting a picture frame is 58 cm long. It is hung over a single hook in the centre of the cord and the cord then makes an angle of 145° as shown in the figure on the right. Calculate the length of the backing of the picture frame, to the nearest centimetre.

111

58 cm 145°

?

Finding angles We can use the cosine rule to find the size of the angles within a triangle. Consider the cosine rule formula. a2 = b2 + c2

2bc cos A

We now make cos A the subject of this formula. a2 = b2 + c2 2bc cos A a + 2bc cos A = b2 + c2 2bc cos A = b2 + c2 a2 b2 + c2 – a2 cos A = ---------------------------2bc 2

In this form, we can use the cosine rule to find the size of an angle if we are given all three side lengths. We should be able to write the cosine rule in three forms depending upon which angle we wish to find. b2 + c2 – a2 cos A = ---------------------------2bc a2 + c2 – b2 cos B = ---------------------------2ac a2 + b2 – c2 cos C = ---------------------------2ab Again, the formula sheet gives the third version of this formula only. The others are an adaptation of the same rule.

WORKED Example 17

A

Find the size of angle B in the triangle on the right, correct to the nearest degree. 7 cm

B

THINK 1

Write the formula with cos B as the subject.

5 cm

9 cm

C

WRITE a2 + c2 – b2 cos B = ---------------------------2ac Continued over page

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Maths Quest General Mathematics HSC Course

THINK

WRITE

2

Substitute a = 9, b = 5 and c = 7.

3

Calculate the value of cos B.

4

Make B the subject of the equation. Calculate B.

5

92 + 72 – 52 cos B = ---------------------------2 9 7 105 cos B = --------126 = 0.8333 B = cos 1(0.8333) B = 34°

equation solver to solve Graphics Calculator tip! Using cosine rule problems (angles) As with all of the trigonometric applications we can use the equation solver to find the required answer. Consider worked example 17. 1. From the MENU select EQUA.

2. Press F3 (SOLV).

3. Delete any existing equation, enter the equation cos B = (92 + 72 – 52) ÷ (2 9 7), and then press EXE .

4. Press F6 (SOLV) to solve the equation.

Your formula sheet will give you two versions of the cosine rule, one for finding a side length and one for finding an angle. When using the equation solver it does not matter which version you use to find a side or an angle. Try using the solver on the equation 52 = 92 + 72 – 2 9 7 cos B. As we found earlier, the cosine ratio for an obtuse angle will be negative. So, when we get a negative result to the calculation for the cosine ratio, this means that the angle we are finding is obtuse. Your calculator will give the obtuse angle when we take the inverse.

Chapter 3 Applications of trigonometry

113

WORKED Example 18 Find the size of angle Q in the triangle on the right, correct to the nearest degree.

Q 4 cm P

THINK

R

6 cm

WRITE

1

Write the formula with cos Q as the subject.

2

Substitute p = 3, q = 6 and r = 4.

3

Calculate the value of cos Q.

4

Make Q the subject of the equation. Calculate Q.

5

3 cm

p2 + r 2 – q2 cos Q = ---------------------------2 pr 32 + 42 – 62 cos Q = ---------------------------2 4 3 – 11 cos Q = --------24 = 0.4583 Q = cos 1( 0.4583) Q = 117°

In some cosine rule questions, you need to work out which angle you need to find. For example, you could be asked to calculate the size of the largest angle in a triangle. To do this you do not need to calculate all three angles. The largest angle in any triangle will be the one opposite the longest side. Similarly, the smallest angle will lie opposite the shortest side.

WORKED Example 19 Find the size of the largest angle in the triangle drawn on the right.

R 3.4 m

4.9 m

S 5.7 m

THINK 1 ST is the longest side, therefore angle R is the largest angle. 2

Write the formula with cos R the subject.

3

Substitute r = 5.7, s = 4.9 and t = 3.4.

4

Calculate the value of cos R.

5

Make R the subject of the equation. Calculate R. Give a written answer.

6 7

T

WRITE

s2 + t 2 – r 2 cos R = -------------------------2st 4.9 2 + 3.4 2 – 5.7 2 cos R = -----------------------------------------2 4.9 3.4 3.08 cos R = ------------33.32 = 0.0924 R = cos 1(0.0924) R = 85° The largest angle in the triangle is 85°.

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Maths Quest General Mathematics HSC Course

Many problems that require you to find an angle are solved using the cosine rule. As always, these begin with a diagram and are finished off by giving a written answer.

WORKED Example 20 Two paths diverge from a point, A. The first path goes for 1.25 km to a point, B. The second path goes for 1.4 km to a point, C. B and C are exactly 2 km apart. Find the angle at which the two paths diverge. THINK 1

WRITE

Draw a diagram.

B

2 km

1.25 km

A 1.4 km 2

Write the formula with cos A as the subject.

3

Substitute a = 2, b = 1.4 and c = 1.25.

4

Calculate the value of cos A.

5

Make A the subject of the equation. Calculate the value of A. Give a written answer.

6 7

b2

c2

C

a2

+ – cos A = ---------------------------2bc

1.4 2 + 1.25 2 – 2 2 cos A = ---------------------------------------2 1.4 1.25 – 0.4775 cos A = ------------------3.5 = 0.1364 A = cos 1( 0.1364) = 98° The roads diverge at an angle of 98°.

remember 1. The cosine rule formulas are: b2 + c2 – a2 • cos A = ---------------------------2bc a2 + c2 – b2 • cos B = ---------------------------2ac a2 + b2 – c2 • cos C = ---------------------------2ab 2. If the value of the cosine ratio is negative, the angle is obtuse. 3. In any triangle, the largest angle lies opposite the largest side and the smallest angle lies opposite the smallest side. 4. Worded problems begin with a diagram and end with a written answer.

115

Chapter 3 Applications of trigonometry

3G

Using the cosine rule to find angles

1 For each of the following, write the cosine rule formula as it applies to the triangle drawn with the boldfaced angle as the subject. a

b P

A

c A

P B

WORKED

Example

17

Q

R

C

M

2 Find the size of the angle marked with the pronumeral in each of the following triangles, correct to the nearest degree. a

b

A

M

θ

θ 8 cm

c

B

11 cm

2.8 m

3.2 m

4.5 m

5.4 m

C B

WORKED

Example

18

C

13 cm

4.0 m

A

θ

N

6.2 m

O

3 In each of the obtuse-angled triangles below find the size of the angle marked with the pronumeral, to the nearest degree. a

b

θ 6m

c 9.6 m

8m

θ 12.9 m

4.2 m α

9.2 m 6.1 m

11 m

4.2 m

4 multiple choice Look at the figure drawn below. 5 cm

3 cm

θ 7 cm

Which of the following correctly represents the value of cos ? A cos

32 + 72 – 52 = ---------------------------2 3 7

B cos

32 + 72 – 52 = ---------------------------2 5 7

C cos

32 + 52 – 72 = ---------------------------2 3 5

D cos

52 + 72 – 32 = ---------------------------2 5 7

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5 multiple choice In which of the following is the angle A

obtuse? B

θ 3 cm

θ 4 cm

3 cm

5 cm

4 cm

4 cm

C

D θ 3 cm

4 cm

3 cm

θ

4 cm

6 cm 4 cm

6 In PQR, p = 7 m, q = 9 m and r = 6 m. Find

QRP, correct to the nearest degree.

7 In KLM, k = 85 mm, l = 145 mm and m = 197 mm. Find the size of the smallest angle, correct to the nearest degree. WORKED

Example

19

8 Calculate the size of all three angles (correct to the nearest degree) in a triangle with side lengths 12 cm, 14 cm and 17 cm. 9 WXYZ is a parallelogram. WX = 9.2 cm and XY = 13.6 cm. The diagonal WY = 14 cm. a Draw a diagram of the parallelogram. b Calculate the size of WXY, correct to the nearest degree.

10 Two roads diverge from a point, P. The first road is 5 km long and leads to a 20 point, Q. The second road is 8 km long and leads to a point, R. The distance between Q and R is 4.6 km. Calculate the angle at which the two roads diverge.

WORKED

Example

11 A soccer goal is 8 m wide. a A player is directly in front of the goal such that he is 12 m from each post. Within what angle must he kick the ball to score a goal? b A second player takes an angled shot. This player is 12 m from the nearest post and 17 m from the far post. Within what angle must this player kick to score a goal? 12 The backing of a picture frame is 50 cm long and is hung over a picture hook by a cord 52 cm long as shown in the figure on the right. Calculate the angle made by the cord at the picture hook.

52 cm

θ

50 cm

Chapter 3 Applications of trigonometry

117

2 1 Find the size of the side marked x, correct to the nearest millimetre.

2 Find the size of the side marked y, correct to 3 significant figures. 4.1 m 11°

y 40° x

346 mm

θ

3 Find the angle marked θ, correct to the nearest degree.

5.8 km

4.9 km

4 Write down the sine rule formula as used to find a side. 5 Use the sine rule to find a, correct to 1 decimal place.

6 Use the sine rule to find θ, correct to the nearest degree.

68°

83°

4.2 km a

θ 57°

7.9 km 14 m

7 Write down the cosine rule formula as used to find a side length. 8 Use the cosine rule to find m, correct to 2 significant figures.

250 m

m

40° 320 m

9 Write down the cosine rule as used to find an angle. 10 Use the cosine rule to find θ, correct to the nearest degree.

9m 13 m

θ 17 m

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Maths Quest General Mathematics HSC Course

Radial surveys In the preliminary course we examined the offset survey. In this survey method an area is measured by drawing a traverse line and measuring offsets at right angles to the traverse line. Because the offset survey created right-angled triangles, the length of each boundary could be calculated using Pythagoras’ theorem and the area could be calculated using the formula Area = 1--- bh. 2 An alternative survey method to this is a radial survey. One type of radial survey is the plane table radial survey. The following steps are taken in a plane table survey. 1. A table is placed in the centre of the field to be surveyed, each corner of the field is sighted and a line is ruled on the paper along the line of sight.

2. The distance from the plane table to each corner is then measured.

25

20 m

m

m

26 m

28

3. The angle between each radial line is then measured and the radial lines joined to complete the diagram.

25

20 m

m

115° 60°

115°

28 m

26 m

70°

The field will then be divided into triangles. The length of each side of the field can then be calculated by using the cosine rule. The perimeter of the field is then found by adding the lengths of each side.

Chapter 3 Applications of trigonometry

WORKED Example 21

119

A 23

30 m

m

The figure on the right is a plane table survey of a block of land. Calculate the perimeter of the block of land, correct to the nearest metre.

B

110° 125° 40°17 m 85°

28 m

C X

D

THINK 1

2

3

4

5

WRITE

Apply the cosine rule in AXB to calculate the length of AB.

Apply the cosine rule in BXC to calculate the length of BC.

Apply the cosine rule in CXD to calculate the length of CD.

Apply the cosine rule in DXA to calculate the length of DA.

Calculate the perimeter by adding the length of each side and rounding the answer to the nearest metre.

For AXB: x2 = a2 + b2 2ab cos X = 302 + 232 2 30 23 = 1900.99 x = 43.6 m The length of AB is 43.6 m. For BXC: x2 = b2 + c2 2bc cos X = 172 + 302 2 17 30 = 407.63 x = 20.2 m The length of BC is 20.2 m. For CXD: x2 = c2 + d 2 2cd cos X = 282 + 172 2 28 17 = 990.03 x = 31.5 m The length of CD is 31.5 m. For DXA: x2 = d2 + a2 2da cos X = 232 + 282 2 23 28 = 2051.77 x = 45.3 m The length of DA is 45.3 m.

cos 110°

cos 40°

cos 85°

cos 125°

Perimeter = 43.6 + 20.2 + 31.5 + 45.3 Perimeter = 140.6 m Perimeter = 141 m (correct to the nearest metre)

A similar approach is used to calculate the area of such a field. The area of each triangle is found using the formula Area = 1--- ab sin C. The total area is then found by 2 adding the area of each triangle.

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Maths Quest General Mathematics HSC Course

WORKED Example 22

A

m 52

Calculate the area of the field on the right. Give your answer correct to the nearest square metre.

m 48

B

96° 144° 120°

67 m

X

C

THINK 1

WRITE

Calculate the area of AXB.

For AXB: Area = 1--- ab sin X =

2

Calculate the area of BXC.

2 1 --2

48

52

sin 96°

= 1241.2 m2 For BXC: Area = 1--- bc sin X =

2 1 --2

67

48

sin 120°

2

3

Calculate the area of CXA.

= 1392.6 m For CXA: Area = 1--- ca sin X =

4

Calculate the total area by adding the area of each triangle.

2 1 --2

52

67

sin 144°

= 1023.9 m2 Total area = 1241.2 + 1392.6 + 1023.9 Total area = 3657.7 m2 Total area = 3658 m2 (correct to the nearest m2)

An alternative to the plane table radial survey is the compass radial survey. In this survey the bearing of each radial line is calculated and this bearing is used to calculate the angle between each radial, as in the worked example below. The method of calculating the perimeter and area of the field is then the same as for the plane table radial survey.

WORKED Example 23

A 338°

B 067°

49

m

m 58

The figure on the right shows a compass radial survey of a block of land. a Calculate the size of AXB. b Hence, calculate the distance AB, correct to the nearest metre.

55

57 m

m X

D 239°

THINK

WRITE

a A is 22° west of North, B is 67° east of North.

a 22° + 67° = 89°

C 114°

Chapter 3 Applications of trigonometry

121

THINK

WRITE

b

b For DAXB: x2 = a2 + b2 - 2ab cos X = 492 + 582 - 2 ¥ 49 ¥ 58 ¥ cos 89° = 5665.8 x = 75 m (correct to the nearest metre) The distance AB is 75 m.

1 2 3 4 5

Write the cosine rule formula. Substitute for a, b and X. Calculate the value of x2. Calculate x. Write your answer.

remember 1. In a radial survey, radial lines are drawn and measured from a point in the centre of an area. 2. In a plane table radial survey, radial lines are drawn on a table by sighting each corner of the field. The length of each line and the angle between the lines is then measured. 3. A compass radial survey is similar but the bearing of each radial line is measured. 4. Each survey divides the area into triangles and the length of each boundary can be calculated using the cosine rule. 5. The area of each triangle can be calculated using the formula Area = 1--- ab sin C. 2

3H

1 The figure on the right is a plane table radial survey of a block of land. Use the cosine 21 rule to calculate the perimeter of the block of land, correct to the nearest metre.

WORKED

Example

15 m

10 m

Radial surveys 100° 70° 80° 110°

m 20

25

m

2 Calculate the perimeter of each of the following areas, correct to the nearest metre.

45 m m

m

115°

80

70° 85° 80° 125° 45

95° 75 m 150°

92 m

m 1

m 114

m 60

55 m

c 12

b 100 m

a

90° 40° 60° 89 m 140° 30°

78

m

122

Maths Quest General Mathematics HSC Course

3 The figure on the right is a plane table survey of a block of land. Calculate the area of the 22 block, correct to the nearest square metre.

WORKED

160° 0m 60° 100° 8 40°

90 m

4 For each of the plane table surveys shown in question 2 calculate the area, correct to the nearest square metre.

A

100 m

11 0m

Example

315°

5 The figure on the right is a compass radial Example survey of a field. 23 a Calculate the size of ∠AXB. b Hence, use the cosine rule to calculate the distance AB, correct to the nearest metre.

B 040°

50

WORKED

40 m

m

60

m X 70 m 110° C

170°

D 350°

100 m

6 Calculate the perimeter of the field given by the compass radial survey on the right. Give your answer correct to the nearest metre.

30 m

90 m

110°

250°

7 Calculate the perimeter of each of the compass radial surveys shown below. a 327° b 339° c 319° 030°

020°

Work

63 m

10 8

m 42

99 m

097°

196°

m 29 226°

085°

m

3.2

49 m

38 m

38

T SHEE

m 72

m 215°

m 114

53 24 m

m

052°

170°

8 For each of the compass radial surveys in question 7 calculate the area, correct to the nearest square metre.

Conducting a radial survey Choose an appropriate area in or near your school to conduct a radial survey. 1 Set up a table in the centre of the area and tape a large piece of paper to the table. 2 Mark a point in the middle of the piece of paper and sight each corner of the field from this point, ruling a line from the point in that direction. 3 Use a tape or trundle wheel to measure the distance from the table to each corner of the field. 4 Use your protractor to measure the angle between each radial line. 5 Calculate the area and the perimeter of the field.

123

Chapter 3 Applications of trigonometry

summary Right-angled triangles • The formulas to be used when solving right-angled triangles are: opposite side sin = ------------------------------hypotenuse adjacent side cos = ------------------------------hypotenuse opposite side tan = ------------------------------adjacent side • To calculate a side length, you need to be given the length of one other side and one angle. • To calculate the size of an angle, you need to be given two side lengths. • If a question is given as a problem, begin by drawing a diagram and give a written answer.

Bearings • Bearings are a measure of direction. • A compass bearing uses the four main points of the compass, north, south, east and west, as well as the four intermediate directions, north-east, north-west, south-east, south-west. • More specific directions are given using true bearings. A true bearing describes a direction as a three-digit angle taken in a clockwise direction from north. • Most bearing questions will require you to draw a diagram to begin the question and require a written answer.

Sine rule • The sine rule allows us to calculate sides and b angles in non-right-angled triangles. • When finding a side length you need to A be given the length of one other side and two angles. A a b c • The sine rule formula is ------------ = ------------ = ------------sin A sin B sin C

C C a B c

B

.

• When finding an angle you need to be given two side lengths and one angle. sin A sin B sin C • The sine rule formula when finding an angle is ------------ = ------------ = ------------- . a b c

Area of a triangle • When you do not know the perpendicular height of a triangle, you can calculate the area using the formula Area = 1--- ab sin C . 2 • To calculate the area using this formula, you need to be given the length of two sides and the included angle.

Maths Quest General Mathematics HSC Course

Cosine rule • The cosine rule allows you to calculate the length of sides and size of angles of non-right-angled triangles where you are unable to use the sine rule. • To find a side length using the cosine rule, you need to be given the length of two sides and the included angle and use the formula c2 = a2 + b2 2ab cos C . • To find an angle using the cosine rule, you need to be given the length of all three a2 + b2 – c2 sides and use the formula cos C = ---------------------------- . 2ab

Surveying • A plane table radial survey sights each corner of a field and draws a radial line in that direction. This divides the field into triangles. The length of each radial line and the angle between radial lines are then measured. • The cosine rule can then be used to calculate the length of each boundary.

25

20

m

m

115° 60°

115°

28

m

26 m

70°

• The formula Area = 1--- ab sin C can be then used to 2

calculate the area of the field. • A compass radial survey takes the bearing of each radial line and this is then used to calculate the angles between them. A 338°

B

m

m 58

067°

49

124

55 D 239°

57 m

m X

C 114°

Chapter 3 Applications of trigonometry

125

CHAPTER review 1 Find the length of the side marked with the pronumeral in each of the right-angled triangles below, correct to 1 decimal place. a b c t 7.9 cm

72° 17.2 cm

3A

42 km 45°

60° x

m

2 In each of the following right-angled triangles, find the size of the angle marked with the pronumeral, correct to the nearest degree. a b c 8.3 km

3A

α 35 cm

16 m

24.8 cm

20.1 km

φ 9m

θ

3 An aeroplane at an altitude of 2500 m sights a ship at an angle of depression of 39°. Calculate, to the nearest metre, the horizontal distance from the aeroplane to the ship.

3A

4 When a yacht is 500 m from shore, the top of a cliff is sighted at an angle of elevation of 12°. a Calculate the height of the cliff, correct to the nearest metre. b Calculate what the angle of elevation of the top of the cliff will be when the yacht is 200 m from shore.

3A

5 Two aircraft are approaching an airport. The Qantas plane (Q) is 40 km due north of the runway (R), while a Jetstar plane (J) is due east of the Qantas plane and north-east of the runway. Calculate the distance of the Jetstar plane from the runway. (Give your answer correct to the nearest metre.)

3B

6 A car rally requires cars to travel for 25 km on a bearing of 240°. The cars are then required to travel due north until they are due west of the starting point. Calculate the distance from the cars to the starting point. (Give your answer correct to 1 decimal place.)

3B

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Maths Quest General Mathematics HSC Course

3B

7 A yacht sails due west for 45 nautical miles before turning north for 23 nautical miles. a Calculate the bearing of the yacht from its starting point. b On what bearing must the yacht sail to return to its starting point?

3C

8 Use the sine rule to calculate each of the sides marked with a pronumeral, correct to 3 significant figures. a b c e a d

19°

117° 70°

28°

31°

4.6 cm

4.6 km

136 mm 20°

3C 3D

9 In XYZ: x = 9.2 cm,

XYZ = 56° and

YXZ = 38°. Find y, correct to 1 decimal place.

10 Use the sine rule to calculate the size of the angle marked with a pronumeral, correct to the nearest degree. a b c α 8 cm

9.7 cm

7.1 m 9°

9 cm

φ

123° 4.1 cm

θ

63°

1.2 m

3E

11 In ABC: b = 46 cm, c = 37 cm and the nearest square centimetre.

3E

12 Find the area of a triangular field with two sides of 80 m and 98 m, which meet at an angle of 130° (correct to the nearest hundred square metres).

3F

13 Use the cosine rule to find each of the following unknown sides, correct to 3 significant figures. a b c 6.9 cm

BAC = 72°. Find the area of the triangle, correct to

6.2 cm

9m

128°

b a

c

5.7 m

50° 117° 11 m

3F

14 In LMN: LM = 63 cm, MN = 84 cm and 1 decimal place.

3F

15 During a stunt show two aeroplanes fly side by side until they suddenly diverge at an angle of 160°. After both planes have flown 500 m what is the distance between the planes, correct to the nearest metre?

4.6 m

LMN = 68°. Find the length of LN, correct to

Chapter 3 Applications of trigonometry

127

16 Use the cosine rule to find the size of the angle in each of the following, correct to the nearest degree. a b c 9 cm 7 cm θ θ

θ

4.2 m

3G

5.3 m

6 cm 15 cm

6 cm 7.9 m 6 cm

17 In XYZ: x = 8.3 m, y = 12.45 m and z = 7.2 m. Find

YZX, to the nearest degree.

18 Two wooden fences are 50 m and 80 m long respectively. Their ends are connected by a barbed wire fence 44 m long. Find the angle at which the two wooden fences meet. 19 The figure below is a plane table radial survey of a field.

3G 3G 3H

60 m m

70 m

30

m

40 80° 120° 50° 110°

a Use the cosine rule to calculate the perimeter of the field. b Calculate the area of the field. 340° 0m 15

260°

160 m

90 m

140 m

20 The figure on the right is a compass radial survey. a Calculate the perimeter of the field. b Calculate the area of the field.

190°

Practice examination questions 1 multiple choice In the figure on the right, which of the following will give the value of x? 13 sin 36° A x = ------------------------sin 64° 13 sin 64° B x = ------------------------sin 36° 13 sin 64° C x = ------------------------sin 80° 13 sin 80° D x = ------------------------sin 64°

13 m 64° x 36°

080°

3H

128

Maths Quest General Mathematics HSC Course

2 multiple choice In the figure on the right, which of the following will give the value of cos ? 62 + 72 – 82 A cos = ---------------------------B cos 2 6 7 C cos

72 + 82 – 62 = ---------------------------2 7 8

D cos

7m

6m

62

82

72

+ – = ---------------------------2 6 8

θ

62 + 72 – 82 = ---------------------------2 7 8

8m

3 multiple choice Maurice walks 3 km on a true bearing of 225°. To return to his starting point he must walk on a compass bearing of: A north-east B north-west C south-east D south-west 4 multiple choice

B 80° 305° A

The figure on the right is a compass radial survey. AXB is: A 35° B 55° C 85° D 135°

X

C 174°

5 The distance between football goal posts is 7 m. If Soon Ho is 20 m from one goal post and 25 m from the other: a draw a diagram showing the goal posts and Soon Ho’s position. b calculate the angle within which Soon Ho must kick to score a goal. (Give your answer correct to the nearest degree.) 6 An observer sights the top of a building at an angle of elevation of 20°. From a point 30 m closer to the building, the angle of elevation is 35° as shown in the figure on 20° the right. A 30 m a Calculate the size of ATB. b Show that the distance BT can be given by the expression 30 sin 20° BT = ------------------------- . sin 15° c Show that the height of the building can be given by the expression 30 sin 20° sin 35° h = -------------------------------------------sin 15° d Calculate the height of the building correct to 1 decimal place.

CHAPTER

3

h 35° B

C

A 345°

110 m

test yourself

7 The figure on the right shows a compass radial survey of a field. a Calculate the length of the boundary CD, correct to 1 decimal place. b Calculate the area of LAXB, correct to the nearest square metre.

T

X 30 m 80 m

D 250°

30 m

B 085°

125° C

Interpreting sets of data

4 syllabus reference Data analysis 5 • Interpreting sets of data

In this chapter 4A Measures of location and spread 4B Skewness 4C Displaying multiple data sets 4D Comparison of data sets

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

4.1

4.2

Finding the mean

1 Find the mean of the following sets of scores. a 3, 5, 8, 3, 9, 4, 3, 5 b Stem Leaf 0 9 1 22367 2 457 3 00

c

Score

Frequency

1

9

2

14

3

32

4

27

5

18

Finding the mode

2 For each of the data sets in question 1 find the mode.

4.3

Finding the median

3 Find the median of the data sets in question 1.

4.4

Finding the range

4 Find each of the data sets in question 1 find the range.

4.5

Finding the interquartile range

5 For each of the data sets in question 1 find the interquartile range.

4.6

Choosing the appropriate standard deviation

6 For each of the following choose and find the appropriate measure of the standard deviation. a At the end of a movie 10 viewers were chosen and asked to rate the movie from 1 to 5 stars. The results were: 3, 4, 2, 3, 1, 5, 2, 4, 3, 5. b At Yass High School there are 80 students who study General Mathematics. They all sat for a test scored out of 20, and the results obtained are given in the table below. Score Frequency

14

15

16

17

18

19

20

3

9

15

24

18

7

4

4.7 Compiling a stem-and-leaf plot

4.9

7 The scores below show the number of customers each day in a car yard. 23, 32, 27, 31, 19, 45, 22, 26, 38, 41, 27, 40, 9, 34, 37, 21, 22, 30, 39, 19, 14, 32, 20, 40, 23, 27, 26, 28, 11, 15, 28, 33 Display the data in a stem-and-leaf plot. Drawing a box-and-whisker plot

8 For the data set in question 6, display the results using a box-and-whisker plot.

Chapter 4 Interpreting sets of data

131

Measures of location and spread Consider the following set of scores that are the exam results for 10 students. 55, 57, 57, 58, 60, 60, 62, 63, 63, 65 To identify a score that is typical in this data set, we can use the mean or median. • The mean is calculated by adding all the scores and dividing by the number of scores in the set. When the data is a small set of scores the mean is found using the formula –x = -----xn where –x = mean x = individual scores (Therefore, x represents the sum of individual scores.) n = number of scores Where the data is presented in a frequency table we use the formula fx –x = ------f where –x = mean x = individual scores f = frequency In this formula fx represents the sum of the frequency score column on the frequency table and f represents the sum of frequency column. • The median is the middle score (odd number of scores) or the average of the two middle scores (even number of scores). For this set of scores: Mean = 600 ÷ 10 = 60 Median = 60 Both the mean and median are a measure of location within a data set.

WORKED Example 1

For the set of scores 13, 19, 31, 40, 55, 65, 90, 92, 95, 100 calculate: a the mean b the median. THINK

WRITE

a

a Total = 600 Mean = 600 ÷ 10 Mean = 60

1 2

Find the total of the scores. Divide the total by the number of scores.

b Average the two middle scores.

b Median = (55 + 65) ÷ 2 = 60

132

Maths Quest General Mathematics HSC Course

We have now examined two data sets. Look at these data sets side by side. Set A: 55, 57, 57, 58, 60, 60, 62, 63, 63, 65 Set B: 13, 19, 31, 40, 55, 65, 90, 92, 95, 100 Although both sets of scores have the same mean and median, they are very different sets of scores. Clearly, in Set B the scores are more spread out than in Set A. To measure the spread of a set of scores, we use one or all of the following. • Range: Highest score lowest score Set A: Range = 65 Set A: Range = 10

55

Set B = 100 Set B = 87

13

• Interquartile range (IQR): The difference between the upper quartile and lower quartile. Set A: Interquartile range = 63 Set A: Interquartile range = 6

57

• Standard deviation: Found using the calculator. Set A:

n

n

Set B: Interquartile range = 92 Set B: Interquartile range = 61

31

(population) or sn (sample) functions on the

= 3.07

Set B:

n

= 31.51

Each of these measures of spread show that in Set B the scores are more scattered than in Set A.

WORKED Example 2

For the set of scores 45, 62, 75, 69, 50, 87, 92 calculate: a the range b the interquartile range c the standard deviation. THINK

WRITE

a Subtract the lowest score from the highest score.

a Range = 92 Range = 47

b

b 45, 50, 62, 69, 75, 87, 92 45, 50, 62, 75, 87, 92

1 2

3

4

Write the scores in ascending order. Divide the data in two halves, leaving the middle score out of both sets. The lower quartile is the median of the lower half; the upper quartile is the median of the upper half. Subtract the lower quartile from the upper quartile.

c Enter the set of scores into your calculator using the statistics function.

45

Lower quartile = 50 Upper quartile = 87

Interquartile range = 87 Interquartile range = 37 c

n

50

= 16.36

Graphics Calculator tip! Finding all summary statistics Your graphics calculator can be used to find all of the important measures of central tendency and spread. This is demonstrated in worked example 3.

Chapter 4 Interpreting sets of data

133

WORKED Example 3 Nadia is a gymnast. For a routine she is given the following scores by 10 judges. 9.0 8.7 9.2 9.3 9.8 9.2 8.8 9.4 9.0 9.1 Use your graphics calculator to find a the mean b the median c the mode d the range e the interquartile range f the population standard deviation g the sample standard deviation. THINK 1

From the MENU select STAT.

2

Delete any existing data, and enter the scores above in List 1.

3

Press F2 (CALC). You may need to first press F6 for more options.

4

Press F6 (SET). Check that 1Var Xlist is set to List 1 and 1Var Freq is set to 1.

5

Press EXE to return to the previous screen, and then press F1 (1Var). All statistics will now be on display using the scroll function.

WRITE

Mean Population standard deviation Sample standard deviation Number of scores Lowest score Lower quartile Median

Upper quartile Highest score Mode

Continued over page

134

Maths Quest General Mathematics HSC Course

THINK a The mean is denoted by the symbol –x .

WRITE a –x = 9.15

b The median is denoted by Med.

b Median = 9.15

c The mode is displayed by Mod. Check the scores for yourself as the data is bimodal only the largest mode is displayed.

c Mode = 9.0 and 9.2

d The range is the highest score (maxX) minus the lowest score (minX).

d Range = 9.8 – 8.7 Range = 1.1

e The interquartile range is the upper quartile (Q3) minus the lower quartile (Q1).

e Interquartile range = 9.3 – 9 Interquartile range = 0.3

f The population standard deviation is denoted by x n.

f

n

g The sample standard deviation is denoted by x n–1.

g

n–1

0.297 0.314

Having identified that the mean and median are measures of location and that range, interquartile range and standard deviation are measures of spread, it is important that you can recognise the effect that the members of a set have on these measures. Consider the case of a basketball team. There are five players on the team, whose heights are: 1.91 m, 1.85 m, 1.52 m, 1.93 m and 1.99 m. The team’s mean height is 1.84 m. Only one of the five players in the team is shorter than the mean height. This is because there is one member of the data set whose height is much less than the others. A score in a data set that is either much less or much greater than all others is called an outlier. An outlier will either reduce or increase the mean such that the mean is no longer typical of the data set. In such cases, the median is a better measure of location than the mean.

Chapter 4 Interpreting sets of data

135

WORKED Example 4

In a small street there are five houses. The values of these houses are: $450 000, $465 000, $465 000, $480 000, $495 000. A new house is built and valued at $750 000. Describe the effect that this outlier has on the: a mean b median c mode (the score that occurs most often). THINK

WRITE

a

a Before new house is built: Total = $2 355 000 Mean = $2 355 000 ÷ 5 = $471 000 After new house is built: Total = $3 105 000 Mean = $3 105 000 ÷ 6 = $517 500 The outlier has caused the mean to increase by $46 500. Only the new house is valued at more than the mean and, as such, has made the mean a poor measure of the typical price.

b

c

1

Calculate the mean before the new house is built.

2

Calculate the mean after the new house is built.

3

Comment on the change in the mean caused by the outlier.

1

Calculate the median before the new house is built.

2

Calculate the median after the new house is built.

3

Comment on the change in the median caused by the outlier.

1

Calculate the mode before the new house is built.

2

Calculate the mode after the new house is built.

After new house is built: Mode = $465 000

3

Comment on the change in the mode caused by the outlier.

The outlier has had no effect on the mode.

b Before new house is built: Median = $465 000 After new house is built: Median = ($465 000 + $480 000) ÷ 2 Median = $472 500 The outlier has caused only a small increase in the median and, as such, the median remains a good measure of the typical score in this data set. c Before new house is built: Mode = $465 000

Generally the mean is the most vulnerable measure of location when an outlier is added to a data set. The median is affected only by the addition of the extra score and is not affected by the size of that score. The outlier will have no effect on the mode.

136

Maths Quest General Mathematics HSC Course

remember 1. The mean and median are measures of location in a data set. • The mean is calculated by adding the scores and then dividing by the number of scores. The mean is calculated using the formulas: –x = Sx –x = Sfx ------------or n Sf • The median is the middle score or the average of the two middle scores in a data set. 2. The range, interquartile range and standard deviation are measures of spread. • The range is the difference between the highest and lowest scores. • The interquartile range is the difference between the upper and lower quartiles. • The standard deviation is found using the sn (population) or sn (sample) functions on the calculator. 3. An outlier is a score in a data set that is either much less or much greater than all other scores in the set. 4. All important summary statistics can be found by entering data into a graphics calculator.

4A SkillS

HEET

4.1

WORKED

Example

1 Finding the mean

SkillS

HEET

4.2 Finding the mode

SkillS

HEET

4.3 Finding the median

EXCE

et

reads L Sp he

One variable statistics

Measures of location and spread

1 The number of goals scored by a team in 10 games of soccer are: 2, 1, 3, 1, 0, 0, 1, 1, 6, 1. a Calculate the mean number of goals scored. b Calculate the median number of goals scored. 2 For each of the following sets of scores, calculate the mean, median and mode (if one exists). a 56, 75, 88, 20, 37, 23, 44 b 2, 1, 7, 4, 6, 1, 1, 4, 5, 3 c 9.9, 9.4, 9.8, 9.6, 9.0, 9.2, 9.8, 9.9 d 13, 15, 16, 17, 10, 13, 15, 14, 19, 20 3 The table at right shows the scores out of 10 by a class of 30 students on a spelling test. a Use the statistics function on your calculator to find the mean score. b Add a cumulative frequency column to the table and use it to calculate the median score. c State the mode.

Score

Frequency

4

2

5

6

6

7

7

9

8

3

9

2

10

1

Chapter 4 Interpreting sets of data

137

Score

Class centre

Frequency 2

6–10

4

11–15

8

16–20

7

21–25

3

26–30

1

UV Stats

program GC

–TI

1–5

Cumulative frequency

am progr –C

asio

GC

4 The table below shows the scores achieved by a football team over a season.

UV Stats

a Copy and complete the table. b Calculate the mean. c Draw a cumulative frequency histogram and polygon and use them to estimate the median. Example

No. of cars

Frequency

6

3

7

5

8

9

9

15

10

11

11

8

12

1

4.5

SkillS

HEET

6 The table below shows the number of cars sold in a car yard each week over one year.

SkillS

HEET

2

5 Below is the number of students in each class at a small primary school. 4.4 28, 29, 27, 28, 30, 28, 25, 27, 23, 28, 27, 28 a Calculate the range of the distribution. Finding b Calculate the interquartile range. the c Use the statistics function on your calculator to find the mean and standard range deviation. Finding the interquartile range

4.6

Skil

HEET

lS a Calculate the range of the number of cars sold. b Add a cumulative frequency column to the table and use the table to calculate: Choosing the i the median appropriate standard ii the upper and lower quartiles deviation iii the interquartile range. L Spre c Use the statistics function on the calculator to find: XCE ad i the mean Boxplots ii the standard deviation. d Draw a box-and-whisker plot of the data.

sheet

E

WORKED

138

Maths Quest General Mathematics HSC Course

7 The table below shows crowds at each match for a team during football season. Crowd

a b c d WORKED

Example

3

Class centre

Frequency

10 000–15 000

5

15 000–20 000

8

20 000–25 000

6

25 000–30 000

4

30 000–35 000

3

Cumulative frequency

Copy and complete the table. Draw a cumulative frequency histogram and polygon. Use the graph in part b to estimate the interquartile range. Find the mean and standard deviation. (Give your answer correct to 2 significant figures.)

8 Below are the scores of two rugby league teams over a period of 10 matches. Team A: 14, 16, 16, 20, 10, 12, 18, 16, 18, 20 Team B: 28, 12, 32, 2, 0, 8, 40, 10, 12, 16 a For each team calculate the mean score. b For each team calculate: i the range ii the interquartile range iii the standard deviation. c Comment on the difference between the performance of the two teams over this 10-game period. The information below is to be used for questions 9 to 12. A basketball squad has eight players. The mean height of the eight players is 1.8 m, and the standard deviation in the heights of the players is 0.1 m. In the first game the tallest player, who is 1.9 m tall, is injured and replaced in the squad by a player who is 1.98 m tall. 9 multiple choice The mean height of the basketball squad will now be: A 1.8 m B 1.81 m C 1.86 m 10 multiple choice As a result of the substitution: A the standard deviation will increase B the standard deviation will decrease C the standard deviation will be unchanged D the effect on the standard deviation cannot be calculated 11 multiple choice As a result of the substitution: A the range will increase B the range will decrease C the range will be unchanged D the effect on the range cannot be calculated

D 1.96 m

Chapter 4 Interpreting sets of data

139

12 multiple choice As a result of the substitution: A the interquartile range will increase B the interquartile range will decrease C the interquartile range will be unchanged D the effect on the interquartile range cannot be calculated 13 James recorded the following five marks on his Maths tests during the year: 78, 77, 80, 85 and 80. 4 a Calculate: i the mean ii the median iii the mode. b In James’ final exam he scored only 20. For the six test results calculate: i the mean ii the median iii the mode. c Describe the effect that the outlier had on the mean, median and mode.

WORKED

Example

14 The mean of a set of five scores is 60. A score of 90 is added to the data set. Describe the effect that this outlier will have on the mean. 15 multiple choice Julie is currently in Year 12. The table below shows the number of days that Julie has been absent from school in each of the previous five years. Year

No. of days absent

7

0

8

1

9

3

10

2

11

0

During Year 12, Julie became seriously ill and was forced to have 37 days off school. According to statistics calculated on Julie’s absences over six years, this outlier will have the greatest effect on: A the mean B the median C the mode D all of the above 16 A small company has four employees who each earn $397.50 per week. Later, a manager is employed who earns $1645.00 per week. a Calculate the mean, median and mode wages. b What effect does the manager’s wage have on the: i mean? ii median? iii mode? c A wage debate is conducted with the employees asking for a rise. Would the mean, median or mode be quoted: i in support of a wage rise by the employees? ii against a wage rise by the employer? Explain your answers.

140

Maths Quest General Mathematics HSC Course

Skewness

8 7 6 5 4 3 2 1 0

Frequency

Frequency

Frequency

By looking at a graph, we can make judgements about the nature of a data set. Consider the first graph shown on the right. This graph is symmetrical and we can see that the mean, median and mode are all equal to 3. The 1 2 3 4 5 majority of scores are clustered around the mean. This is an example of a normal distribution. We can compare the standard deviation of data 6 sets by looking at such graphs. The more clustered 5 the data set, the smaller the standard deviation. 4 The second graph is still normally distributed 3 with the mean, median and mode still equal to 3. 2 1 However, there are more scores which are further 0 away from the mean and, hence, the standard 1 2 3 4 5 deviation of the data set is greater. The third graph shows a data set where the scores are not clustered and there are two modes 6 at either end of the distribution. 5 In this example, although it is still symmetrical 4 there are two modes, 1 and 5, while the mean and 3 2 median are still 3. The standard deviation in this 1 distribution is greater than either of the two pre0 vious examples as there are more scores further 1 2 3 4 5 away from the mean. The mean and median can be seen from the graph only because it is symmetrical.

The figure on the right shows the distribution of a set of scores on a spelling test. a Is the graph symmetrical? b What is the mode(s)? c Can the mean and median be seen from the graph?

Frequency

WORKED Example 5 5 4 3 2 1 0 6 7 8 9 10 Score

THINK

WRITE

a The columns either side of the middle are equal.

a The graph is symmetrical.

b The scores that occur the most often are 7 and 9.

b Mode = 7 and 9

c The middle score will be the mean and median.

c Mean = 8, median = 8

Chapter 4 Interpreting sets of data

141

8 7 6 5 4 3 2 1 0

Frequency

Frequency

When a graph is not symmetrical, the mean and median cannot be easily seen from the graph. Consider the distribution in the graph below left. The way in which the data are gathered to one end of the distribution is called the skewness. A greater number of scores are distributed at the lower end of the distribution. In this case, the data are said to be positively skewed. Similarly, when most of the scores are distributed at the upper end, the data are said to be negatively skewed, as shown in the graph below right.

1 2 3 4 5

8 7 6 5 4 3 2 1 0

1 2 3 4 5

20 16 12 8 4 0 51 –6 61 0 –7 71 0 – 81 80 91 –90 –1 00

The distribution on the right shows the results of the Maths Trial HSC at a certain school. a What is the modal class? b Describe the skewness of the data set shown on the right.

Frequency

WORKED Example 6

Maths results

THINK

WRITE

a The class occurring the most often is the 81–90 class.

a Modal class = 81

b The majority of data are at the upper end of the distribution.

b The data are negatively skewed.

90

remember 1. A distribution is symmetrical when the data are equally distributed around the mean. 2. When the data are symmetrical, the median and mean will both be the middle score. 3. When the data are clustered around the mean, the standard deviation is smaller. 4. When the majority of scores are at the lower end of a distribution, it is said to be positively skewed. 5. When the majority of scores are at the upper end of the distribution, it is said to be negatively skewed.

Maths Quest General Mathematics HSC Course

4B WORKED

Example

5

Skewness Frequency

142

1 In the distribution on the right: a is the graph symmetrical? b what is the modal class(es)? c can the mean and median be seen from the graph? and, if so, what are their values?

12 10 8 6 4 2 0 1 2 3 4 5 7 6 5 4 3 2 1 0 0– 4 5– 10 9 – 15 14 –1 20 9 – 25 24 –2 9

Frequency

2 For the distribution shown on the right: a are the data symmetrical? b what is the modal class(es)? c can the mean and median be seen from the graph? and, if so, what are their values?

Example

6

Frequency

0

6

1

4

2

4

3

4

4

4

5

6

4 For the distribution shown on the right: a what is the modal class(es)? b describe the skewness of the distribution.

12 10 8 6 4 2 0 0– 1 1– 2 2– 3 3– 4 4– 5

WORKED

No. of goals

Frequency

3 The table on the right shows the number of goals scored by a hockey team throughout a season. a Show this information in a frequency histogram. b Are the data symmetrical? c What is the mode(s)? d Can the mean and median be seen for this distribution? and, if so, what are their values?

5 For each of the following dot plots describe the skewness of the distribution. a b c •

• •

•• •

•• ••

•• ••

0 1 2 3 4 5

• •• • •• • •• • 14 15 16 17 18 19 20

•• • •

•• •• • • •

9.5 9.6 9.7 9.8 9.9 10

6 For the stem-and-leaf plots drawn below describe the distribution a Key 3|5 = 35 b Key 4|3 = 4.3 4*|6 = 4.6 Stem Leaf Stem Leaf 2 259 2* 9 3 0012589 3* 0 4 4 2289 3* 5 5 8 8 5 09 4* 0 0 0 1 1 3 4 4 6 0 4* 5 5 5 6 7 8 8 9 9

Chapter 4 Interpreting sets of data

7 The table below shows the number of goals scored by a basketball team throughout a season. No. of goals

Frequency

11–20

3

21–30

6

31–40

7

41–50

23

51–60

21

a Draw a frequency histogram of the data. b Describe the data set in terms of its skewness. 8 multiple choice Which of the distributions below has the smallest standard deviation?

Frequency

C

B Frequency

10 8 6 4 2 0

1 23 4 5

6 5 4 3 2 1 0

12345

D

6 5 4 3 2 1 0 1 23 4 5

Frequency

Frequency

A

8 7 6 5 4 3 2 1 0 1 2 3 4 5

The distribution represented by the graph on the right is: A positively skewed B negatively skewed C symmetrical D normally distributed

Frequency

9 multiple choice 16 14 12 10 8 6 4 2 0

1 2 3 4 5

143

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10 A movie is shown at a cinema 30 times during the week. The number of people attending each session of the movie is shown in the table below.

a b c d

No. of people

Frequency

1–50

2

51–100

3

101–150

5

151–200

10

201–250

10

Present the data in a frequency histogram. Are the data symmetrical? What is the modal class(es)? Describe the skewness of the distribution.

11 Year 12 at Wallarwella High School sit exams in Chemistry and Maths. The results are shown in the table below. Mark

Chemistry

Maths

31–40

2

3

41–50

9

4

51–60

7

6

61–70

4

7

71–80

7

9

81–90

9

7

91–100

2

4

Work

a Is either distribution symmetrical? b If either distribution is not symmetrical, state whether it is positively or negatively skewed. c State the mode of each distribution. d In which subject is the standard deviation greater? Explain your answer. T SHEE

4.1

12 Draw an example of a graph which is: a symmetrical b positively skewed with one mode c negatively skewed with two modes.

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1 Consider the following set of scores: 23, 45, 24, 19, 22, 16, 16, 27, 20, 21. Calculate the following measures of location and spread. 1 Mean 2 Median 3 Mode 4 Range 5 Interquartile range 6 Standard deviation 7 For the data set, describe the skewness of the distribution. 8 Does the data set have an outlier? 9 Which measure of central tendency is the best measure of location in this data set? 10 Explain why the interquartile range is a better measure of spread than the range.

Displaying multiple data sets Two data sets can be compared using a number of the displays that have been studied in earlier parts of this course. Presenting both sets of data on the same display gives a quick and easy comparison.

Stem-and-leaf plots Two sets of data can be displayed on the same stem-and-leaf plot. This is done by having the stem in the centre of the plot, with both sets of data back to back.

WORKED Example 7

The data shown below display the marks of 15 students in both English and Maths. English: 45 67 81 59 66 61 78 71 74 91 60 49 58 62 70 Maths: 85 71 49 66 64 68 75 71 69 60 63 80 87 54 59 Display the data in a back-to-back stem-and-leaf plot. THINK 1 2 3 4

WRITE

Write a key at the top of the stem-and-leaf plot. Draw the stem showing categories of 10 in the centre of the page. Display the information for English on the left of the stem. Display the information for Maths on the right of the stem.

Key: 4 5 = 45 English 95 4 98 5 76210 6 8410 7 1 8 1 9

Maths 9 49 034689 115 057

This stem-and-leaf plot allows for both distributions to be easily seen, and for a judgement on the skewness of the distribution to be made.

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Box-and-whisker plots A single scale can be used. Draw two box-and-whisker plots on that scale that will allow the comparison of the median, range and interquartile range of two distributions to be compared.

WORKED Example 8 Use the back-to-back stem-and-leaf plot drawn in worked example 7 to: a calculate the median of each distribution b calculate the range of each distribution c calculate the interquartile range of each distribution d draw a box-and-whisker plot of each distribution on the same scale. THINK

WRITE

a The median will be the eighth score in each distribution.

a English median = 66 Maths median = 68

b To calculate the range of each distribution, subtract the lowest score from the highest score.

b English range = 91 45 English range = 46 Maths range = 87 49 Maths range = 38

c

The lower quartile will be the fourth score. The upper quartile will be the twelfth score. The interquartile range is the difference between the quartiles.

c English lower quartile = 59 Maths lower quartile = 60 English upper quartile = 74 Maths upper quartile = 75 English interquartile range = 74 59 English interquartile range = 15 Maths interquartile range = 75 60 Maths interquartile range = 15

Draw a scale. Draw the English box-and-whisker plot. Draw the Maths box-and-whisker plot.

d

1 2 3

d

1 2 3

English Maths

0 10 20 30 40 50 60 70 80 90 100 Scale

Examining exam results Collect data on the most recent exam that has been done in your class. 1 Display the data in a stem-and-leaf plot. 2 Find all the information needed to display the data in a box-and-whisker plot. 3 Is there any skewness evident in the data? 4 Which measure of location best describes the typical score in this data set?

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Graphics Calculator tip! Storing multiple data sets Your graphics calculator can store two sets of data at the same time. Consider worked example 8, which uses the data from worked example 7. By graphing each data set using a box-and-whisker-plot we can also easily retrieve the rest of the required information by using the trace function. 1. From the MENU select STAT.

2. Enter the data for English in List 1 and the data for Maths in List 2.

3. Press F1 (GRPH), and then F6 (SET). We will set the English data as GPH1 and the Maths data as GPH2. Press F1 (GPH1) and enter the settings shown in the screen at right.

4. Press EXE once these settings have been entered. Press F6 (SET) and F2 (GPH2), and again enter the settings shown at right.

5. To show both graphs on the same screen press EXE after entering the settings above, press F4 (SEL), and set both StatGraph1 and StatGraph2 to DrawOn as shown.

6. Press F6 (DRAW) to draw both graphs.

7. Press SHIFT F1 (TRACE), and use the arrow keys to move around to the five key points on each graph. The screen at right displays the median for the Maths data.

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Multiple sets of data can be displayed on the same set of axes for many different graphs. This is particularly useful when comparing data suitable for display on a radar chart.

WORKED Example 9

The table below shows the number of admissions to two hospitals, each month, over a one-year period. Display both sets of data on a radar chart. Month

Hospital A

Hospital B

January

3

15

February

6

12

March

7

9

April

9

10

May

10

8

June

15

7

July

14

9

August

16

6

September

10

8

October

5

5

November

3

9

December

7

2

THINK 1 2 3

Draw the radar with a 30° angle between the months. Draw a scale around the radar. Plot each set of points.

WRITE Hospital A

Jan Dec 20 15 Nov 10 5 Oct 0

Hospital B

Feb March April

Sep

May Aug

July

June

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Area charts are another method of comparing information. In an area chart, line graphs are stacked on top of each other, thus allowing the area between each line graph to serve as the comparison between the data sets.

WORKED Example 10 The table below shows the amount of rainfall, in millimetres, in Sydney, Melbourne and Brisbane each month throughout a year.

Sydney Melbourne Brisbane

January

February

March

April

May

June

103

117.1

133.7

126.6

120.4

131.7

49

47.7

51.8

58.4

57.2

50.2

159.6

158.3

140.7

92.5

73.7

67.8

July

August

September

October

November

December

Sydney

98.2

79.8

69.9

77.5

83.1

79.6

Melbourne

48.7

50.6

59.4

67.7

60.2

59.9

Brisbane

56.5

45.9

45.7

75.4

97

133.3

Show this information in an area chart.

2

3

4

Draw a pair of axes. The vertical axis will need to be at least the rainfall total of all three cities in the wettest month. Draw a line graph of Sydney’s rainfall, and shade the area below it. Add Melbourne’s rainfall to Sydney’s rainfall, and draw a line graph showing these figures. Colour the area between the two graphs, as this area represents Melbourne’s rainfall. Add Brisbane’s rainfall to the previous total. Colour the area above the previous line, as this area represents Brisbane’s rainfall.

Brisbane Rainfall (mm)

1

WRITE

Melbourne

Sydney

350 300 250 200 150 100 50 0 Ja n Fe b M a Apr Mr ay Ju n Ju Au l g Se p Oc No t v De c

THINK

remember 1. Data can be compared by showing two sets of data on the same display. 2. Two sets of data are shown on a stem-and-leaf plot by displaying the data back to back. 3. Two box-and-whisker plots can be drawn on the same scale to compare the ranges, interquartile ranges and medians. 4. A radar chart can be used to compare trends over a period of time by plotting two sets of data on one radar chart. 5. An area graph can be used to compare multiple sets of data. The area in each section of the graph allows for comparison between quantities.

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SkillS

4C HEET

4.7

WORKED

Example

7

SkillS

Compiling a stem-andleaf plot

HEET

4.8

SkillS

4.9

WORKED

Example

Drawing a box-andwhisker plot

1 In a class of 30 students there are 15 boys and 15 girls. Their heights are measured and are listed below. Boys: 1.65, 1.71, 1.59, 1.74, 1.66, 1.69, 1.72, 1.66, 1.65, 1.64, 1.68, 1.74, 1.57, 1.59, 1.60 Girls: 1.66, 1.69, 1.58, 1.55, 1.51, 1.56, 1.64, 1.69, 1.70, 1.57, 1.52, 1.58, 1.64, 1.68, 1.67 Display this information in a back-to-back stem-and-leaf plot. 2 The number of points scored in each match by two rugby union teams are shown below. Team 1: 34, 32, 24, 25, 8, 18, 17, 23, 29, 40, 19, 42 Team 2: 23, 20, 35, 21, 46, 7, 9, 24, 27, 38, 41, 30 Display these data in a back-to-back stem-and-leaf plot.

Finding the mean, median, mode from a stem-and-leaf plot

HEET

Displaying multiple data sets

8

3 The stem-and-leaf plot below is used to display the number of vehicles sold by the Ford and Holden dealerships in a Sydney suburb each week for a three-month period. Key: 1 5 = 15 Ford Holden 74 0 39 952210 1 111668 8544 2 2279 0 3 5 a State the median of both distributions. b Calculate the range of both distributions. c Calculate the interquartile range of both distributions. d Show both distributions on a box-and-whisker plot. 4 A motoring organisation tests two different brands of tyres. Twenty tyres of each brand are tested to find out the number of kilometres each tyre could travel before the tread had worn down. The results are shown in the stem-and-leaf plot below. Key: 1 2 = 12 000 km 1* 7 = 17 000 km Brand A Brand B 9 8 0* 43110 1 0011224 7 7 7 6 6 5 1* 5 6 7 8 8 8 9 4431100 2 0134 2* 5 5 Draw two box-and-whisker plots on the same scale to display this information. 5 The figures below show the ratings of two radio stations each week over a threemonth period. Station A: 9.2, 9.4, 9.2, 9.5, 9.7, 9.9, 10.1, 9.1, 8.8, 8.7, 9.0, 8.5, 9.3 Station B: 8.5, 8.1, 8.2, 8.9, 9.0, 9.2, 8.4, 8.7, 8.8, 10.5, 11.2, 11.4, 8.7 a Display the information in a back-to-back stem-and-leaf plot. b Use the stem-and-leaf plot to display both sets of data on the same box-andwhisker plot.

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6 The box-and-whisker plot drawn on the Team A right displays statistical data for two AFL Team B teams over a season. 50 60 70 80 90 100110120130140 150 160 Scale a Which team had the higher median score? b What was the range of scores for each team? c For each team calculate the interquartile range. 7 The two five-number summaries below show the performance of Emad and Larry on their Mathematics exams throughout the year. Emad: 45, 64, 68, 76, 80 Larry: 51, 58, 65, 72, 75 a Compare the performance of Emad and Larry on a box-and-whisker plot. b What is the range for both students? c What is the interquartile range for both students? 8 multiple choice The box-and-whisker plot drawn on the right Physics shows Emma’s performance in her Physics Chemistry and Chemistry exams. Which of the following 0 10 20 30 40 50 60 70 80 90 100 Scale statements is correct? A The median of Emma’s mark in Physics is greater than for Chemistry. B The range of Emma’s marks in Physics is greater than in Chemistry. C The interquartile range of Emma’s marks in Physics is greater than in Chemistry. D All of the above. WORKED

Example

9

9 This radar chart shows the average daily maximum temperature for both Sydney and Melbourne for each month of a year. a Which month had the lowest temperature in Sydney? b What was the range of temperatures in Melbourne? c What was the average of the temperatures in Sydney?

Sydney temperature (°C) Melbourne temperature (°C) J F D 30

A

S

M A

WORKED

Brisbane

J

2 pm

8 am 12 noon

Melbourne

10 am

Sydney

40 35 30 25 20 15 10 5 0

Jan Fe b Ma Apr Ma r y Jun Ju Au l g Se p Oc No t Dev c

Average no. of rainy days per month

J

Supermarket X Supermarket Y 12 midnight 2 am 10 pm 120 100 80 60 8 pm 4 am 40 20 0 6 pm 6 am 4 pm

11 This area chart shows the average number of rainy days each month in Sydney, Melbourne and Brisbane. Display 10 this information as a table.

M

10 0

O

10 This radar chart shows the number of customers in two different supermarkets at two-hour intervals. a Find the range for each supermarket. b Describe the general pattern at each supermarket.

Example

20

N

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Computer Application 1 Displaying statistical data EXCE

et

reads L Sp he

1. From the Maths Quest General Mathematics HSC Course CD-ROM, access the spreadsheet Fast Food Sales.

Fast Food Sales

2. In cell B12 use the spreadsheet’s inbuilt statistical function to find McDonald’s average daily sales. [=AVERAGE(B4:B10)] 3. In cell B13 use the spreadsheet’s inbuilt statistical function to find the standard deviation of McDonald’s daily sales. [=STDEV(B4:B10)] 4. Under Edit, use the Fill and Right functions to copy these formulas for KFC and Pizza Hut. 5. Use the charting facility to draw an area chart of the figures presented.

Comparison of data sets When multiple data displays are used to display similar sets of data, comparisons and conclusions can then be drawn about the data. Multiple displays such as stem-and-leaf plots and box-and-whisker plots allow for comparison of statistics such as the median, range and interquartile range, while radar charts and area charts allow for trends and overall quantities to be compared.

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WORKED Example 11

A bank surveys the average morning and Key: 1 2 = 1.2 minutes Morning Afternoon afternoon waiting time for customers. 7 0 788 The figures were taken each Monday to 86311 1 1124456667 Friday in the morning and afternoon for 9 6 6 6 5 5 4331 2 2558 one month. The stem-and-leaf plot at right 952 3 16 shows the results. 5 4 a Find the median morning waiting time 5 7 and the median afternoon waiting time. b Calculate the range for morning waiting times and the range for afternoon waiting times. c What conclusions can be made from the display about the average waiting time at the bank in the morning compared with the afternoon? THINK a There are 20 scores in each set and so the median will be the average of the 10th and 11th scores.

WRITE a Morning:

b For each data set, subtract the lowest score from the highest score.

b Morning:

Afternoon:

Afternoon: c Conclude that waiting time in the afternoon is generally less and more consistent except for one outlier.

Median = (2.4 + 2.5) ÷ 2 = 2.45 minutes Median = (1.6 + 1.6) ÷ 2 = 1.6 minutes Range = 4.5 0.7 = 3.8 minutes Range = 5.7 0.7 = 5 minutes

c The waiting time is generally shorter in the afternoon. There is one outlier in the afternoon scores which causes the range to be larger. However, apart from this outlier the afternoon scores are less spread.

Two-way tables can also be a meaningful way of displaying data. A two-way table allows for two variables to be compared.

WORKED Example 12

A survey of 25 000 people is taken. The sex of each respondant is noted and whether they are a smoker or non-smoker is also noted. The results are displayed in the two-way table below. Males

Females

Totals

Smokers

4 125

4 592

8 717

Non-smokers

8 436

7 847

16 283

12 561

12 439

25 000

Totals

a What percentage of the females surveyed were smokers? b What percentage of the smokers surveyed were female? Continued over page

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THINK

WRITE

a Write 4592 as a percentage of 12 439.

a Percentage of females who smoke 4592 = ---------------- 100% 12 439 = 36.9%

b Write 4592 as a percentage of 8717.

b Percentage of smokers who are female 4592 = ------------ 100% 8717 = 52.7%

The most common method, however, for comparing data sets is to compare the summary statistics from the data sets. The measures of location such as mean and median are used to compare the typical score in a data set. Measures of spread such as range, interquartile range and standard deviation are used to make assessments about the consistency of scores in the data set.

WORKED Example 13 Below are the scores for two students in eight Mathematics tests throughout the year. Jane: 45, 62, 64, 55, 58, 51, 59, 62 Pierre: 84, 37, 45, 80, 74, 44, 46, 50 a Use the statistics function on the calculator to find the mean and standard deviation for each student. b Which student had the better overall performance on the eight tests? c Which student was more consistent over the eight tests? THINK

WRITE

a Enter the statistics into your calculator and use the x function for the mean and the n function for the standard deviation.

a Jane: x = 57, n = 6 Pierre: x = 57.5, n = 17.4

b The student with the higher mean performed better overall.

b Pierre performed slightly better overall, as his mean mark was higher than Jane’s.

c The student with the lower standard deviation was more consistent.

c Jane was the more consistent student, as her standard deviation was much lower than Pierre’s.

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remember 1. When multiple displays are used for two or more sets of data, we can compare and contrast the data sets and determine if any relationship exists between them. 2. A multiple stem-and-leaf plot allows for a quick comparison of medians, ranges and interquartile ranges. 3. The summary statistics from two data sets can be compared quickly on a box-and-whisker plot. 4. Two-way tables can be used to make a comparison of data where two variables are involved. 5. The most commonly used comparisons are summary statistics to compare what is a typical score and what the spread of the data is.

4D WORKED

Example

11

Comparison of data sets

1 The stem-and-leaf plot drawn below shows the marks obtained by 20 students in both English and Maths. Key: 7 1 = 71 English 7410 9976653110 87752 2

4 5 6 7 8 9

Maths 17 24799 133466 4448 36 4

a Calculate the median mark for both English and Maths. b Calculate the range of marks for both English and Maths. c Comment on the distribution of marks in each of the subjects. 2 Tracey measures the heights of twenty Year 10 boys and twenty Year 10 girls and produces the following five-number summaries for each data set. Boys: 1.47, 1.58, 1.64, 1.72, 1.81 Girls: 1.55, 1.59, 1.62, 1.66, 1.73 a Draw a box-and-whisker plot for both sets of data and display them on the same scale. b What is the median of each distribution? c What is the range of each distribution? d What is the interquartile range for each distribution? e Comment on the spread of the heights among the boys and the girls.

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3 The box-and-whisker plots on the right show the heights of a sample of Year 7 boys and a similar-sized sample of Year 12 boys.

Year 7 Year 12 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 m Scale

a Calculate the range of heights among both the Year 7 and Year 12 boys. b Calculate the interquartile range of the heights among both the Year 7 and Year 12 boys. c Comment on the relationship between the two data sets, both in terms of measures of location and measures of spread. 4 The values of hardware and software sales for a chain of computer stores are shown for each month in the radar chart on the right. Comment on any relationship observed in this chart between the sales of hardware and the sales of software.

Hardware

N O

D 2.5 2 1.5 1 0.5 0

J

Software

F M A

S

M A

WORKED

Example

12

1200

J

Western region

1000 Rainfall (mm)

5 The area chart on the right shows the rainfall in four areas of New South Wales throughout the year. a Which region has the greatest rainfall? b In which region is the range of rainfall figures least? c What relationship exists between the rainfall in each of the areas?

J

Southern region

800 600

North/Eastern region

400

North/Western region

200

0 mer utumn Winter Spring m A Su

6 The two-way table below shows the results of random breath testing by Sydney police over one weekend. A driver is charged if they record a reading of 0.05% prescribed concentration of alcohol (PCA). Males

a b c d e

Females

Totals

Over 0.05 PCA

26

7

33

Below 0.05 PCA

962

743

1705

Totals

988

750

1738

What percentage of the drivers tested were female? What percentage of the drivers tested had a PCA over 0.05? What percentage of female drivers had a PCA over 0.05? What percentage of male drivers had a PCA over 0.05? Based on the above results, can any conclusion be drawn concerning the prevalence of drink driving among males and females? Explain your answer.

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7 Ashley is the star player of a football team. To analyse the importance of Ashley to the team, the coach prepares the two-way table below showing the results of games over three years both when Ashley is playing and not playing. Won

Lost

Totals

Ashley playing

38

4

42

Ashley not playing

10

8

18

Totals

48

12

60

a What percentage of games were won when Ashley played? b What percentage of games were won when Ashley did not play? c Do you think that Ashley has a significant impact on the performance of the team? Explain your answer. 8 To compare the performance of city and country students in the HSC, the number of students achieving a UAI of at least 90 is studied in six city and six country high schools. City

Country

Totals

90

58

61

119

UAI < 90

551

569

1120

Totals

609

630

1239

UAI

a What percentage of city students achieved a UAI of at least 90? b What percentage of country students achieved a UAI of at least 90? c Of those students who achieved a UAI of at least 90, what percentage were from: i the city? ii the country? d Based on the above results, could any conclusion be drawn about the performance of city and country students in the HSC? WORKED

Example

13

9 Calvin recorded his marks for each test that he did in Physics and Chemistry throughout the year. Physics: 65, 74, 69, 66, 72, 64, 75, 60 Chemistry: 45, 85, 91, 42, 47, 72, 87, 85 a In which subject did Calvin achieve the better average mark? b In which subject was Calvin more consistent? Explain your answer. 10 The police set up two radar speed checks in a country town. In both places the speed limit is 60 km/h. The results of the first 10 cars that have their speed checked are given below. Point A: 60, 62, 58, 55, 59, 56, 65, 70, 61, 64 Point B: 55, 58, 59, 50, 40, 90, 54, 62, 60, 60 a Calculate the mean and standard deviation of the readings taken at each point. b At which point are drivers generally driving faster? c At which point is the spread of the readings taken greater? Explain your answer.

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11 Aaron and Sunil open the batting for the local cricket team. The number of runs they have scored in each innings this season are listed below. Aaron: 45, 43, 33, 56, 21, 38, 0, 29, 76, 40 Sunil: 5, 70, 12, 54, 68, 11, 8, 64, 32, 69 a Calculate the mean number of runs scored for each player. b What is the range of runs scored by each player? c What is the interquartile range of runs scored by each player? d Which player would you consider to be the more consistent player? Explain your answer. 12 multiple choice Andrea surveys the age of people attending a concert given by two bands. The boxand-whisker plot shown below shows the results. Band A Band B

0 10 20 30 40 50 60 70 80 Scale

Which of the following conclusions could be drawn based on the above information? A Band A attracts an older audience than Band B. B Band A appeals to a wider age group than Band B. C Band B attracts an older audience than Band A. D None of the above. 13 multiple choice Two drugs are tested to see which is more effective at fighting disease. The results are displayed in the two-way table below.

Recovered Not recovered Totals

Drug 1

Drug 2

Totals

124

136

260

32

45

77

156

181

337

Of those patients who recovered, the percentage who were treated with drug 1 was: A 46.3% B 47.7% C 69.0% D 79.5%

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14 multiple choice

15 A company producing matches advertise that there are 50 matches in each box. Two machines are used to distribute the matches into the boxes. The results from a sample taken from each machine are shown in the stemand-leaf plot below. Key: 5 1 = 51 5* 6 = 56 Machine A Machine B 4 4 9 9 8 7 7 6 6 5 4* 5 7 8 9 9 9 9 9 9 9 9 43222211100000 5 0000011111223 5 5 5* 9 a Display the data from both machines on a box-and-whisker plot. b Calculate the mean and standard deviation of the number of matches distributed from both machines. c Which machine is the more dependable? Explain your answer.

Developing a two-way table Conduct a survey to determine the number of Year 12 students at your school who have their driver’s licence. Use the results to complete the two-way table below. Males

Females

Totals

Licensed Unlicensed Totals a b c d

What percentage of females have their driver’s licence? What percentage of licensed drivers are female? Are these figures the same? Design a data set where the percentage of females with their licence is equal to the percentage of licensed drivers who are female. e What conditions are necessary for these two percentages to be equal?

Work

The figures below show the ages of the men’s and women’s champions at a tennis tournament. Men’s: 23, 24, 25, 26, 25, 25, 22, 23, 30, 24 Women’s: 19, 27, 20, 26, 30, 18, 28, 25, 28, 22 Which of the following statements is correct? A The mean age of the men’s champions is greater than the mean age of the women’s champions. B The range is greater among the men’s champions than among the women’s champions. C The interquartile range is greater among the men’s champions than among the women’s champions. D The standard deviation is greater among the men’s champions than among the women’s champions.

T SHEE

4.2

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summary Measures of location and spread • Measures of location give the typical score in the data set. The mean, median and mode are measures of location. • The mean of a small data set is found using: –x = -----xn – where x = the mean, x = individual scores and n = number of scores. • Where data is in a table, the mean is found using: fx –x = ------f where –x = the mean, x = individual scores and f = frequency. • Measures of spread describe how spread out the data are. The range, interquartile range and standard deviation are measures of spread. • An outlier is a single score that is much greater or much less than most of the scores. The outlier may have a great effect on the mean but has only a slight effect on the median and no effect on the mode in a small data set. The larger the data set, the less the effect a single outlier will have.

Skewness • When the data are symmetrical, they are said to be normally distributed. • The more clustered the data are around the mean, the smaller the standard deviation. • When most of the data are below the mean, the data are said to be positively skewed. • When most of the data are above the mean, the data are said to be negatively skewed.

Displaying multiple data sets • Two sets of data can be displayed on a stem-and-leaf plot by displaying the data back to back. • The summary statistics from two data sets can be displayed by using the same scale and drawing two box-and-whisker plots. • Two sets of data can be displayed on a radar chart to display related trends over a period of time. • An area chart can be drawn to display several sets of data. The area in each section of the graph then displays the quantities for comparison.

Comparison of data sets • The summary statistics from two data sets can be compared from a stem-and-leaf plot or box-and-whisker plot. • Two-way tables are used to compare data where there are two variables involved. • Data are most commonly compared using the mean and standard deviation.

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CHAPTER review 1 Below are the ages of 15 players in a soccer squad. 23, 28, 25, 19, 17, 28, 29, 29, 22, 21, 35, 30, 22, 27, 26 a Calculate the mean age of the players in the squad. b Find the median age of players in the squad.

4A

2 The table below shows the number of house calls that a doctor has been required to make each day over a 32-day period.

4A

a b c d

Number of house calls

Frequency

0

1

1

6

2

8

3

9

4

6

5

2

Copy the table into your workbook and add a cumulative frequency column. Calculate the mean number of house calls per day. Find the median number of house calls per day. What is the modal number of house calls per day?

3 The set of figures shown below shows the number of pages in a daily newspaper every day for two weeks. 72, 68, 76, 80, 64, 60, 132, 72, 84, 88, 60, 56, 76, 140 a What is the mean number of pages in the newspaper? b What is the range? c What is the interquartile range? d Use the statistics function on your calculator to find the standard deviation.

4A

4 The table below shows the number of rescues that are made each weekend at a major beach.

4A

Number of rescues

Frequency

8

2

9

5

10

12

11

3

12

0

13

1

14

3

Use the statistics function on your calculator to find the mean and the standard deviation of these data.

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Maths Quest General Mathematics HSC Course

5 The table below shows the customer waiting time at 10 am each morning at a bank over an 8-week period. Waiting time

a b c d

Class centre

Frequency

0–1 minute

1

1–2 minutes

4

2–3 minutes

10

3–4 minutes

13

4–5 minutes

9

5–6 minutes

3

Cumulative frequency

Copy and complete the table. Use the statistics function on your calculator to find the mean and standard deviation. Draw a cumulative frequency histogram and polygon. Use the graph to estimate the interquartile range of the data.

6 The figures below show the number of points scored by a basketball player in six matches of a tournament. 36, 2, 38, 41, 27, 33 a Calculate the mean number of points per game. b Calculate the median number of points per game. c Explain why there is such a large difference between the mean and the median.

4B

7 Consider the data set represented by the frequency histogram on the right. a Are the data symmetrical? b Can the mean and median of the data be seen? c What is the mode of the data?

4B

8 The table below shows the number of attempts that 20 members of a Year 12 class took to obtain a driver’s licence. Number of attempts

Frequency

1

11

2

4

3

2

4

2

5

0

6

1

Frequency

4A

8 7 6 5 4 3 2 1 0

15 16 17 18 19 20

a Show these data in a frequency histogram. b Are the data positively or negatively skewed?

4B

9 Draw an example of a frequency histogram for which the data are negatively skewed.

Chapter 4 Interpreting sets of data

163

10 The figures below show the marks obtained by 20 students in English and Maths. English: 56, 45, 57, 56, 65, 82, 74, 80, 91, 84, 68, 52, 67, 64, 60, 66, 74, 77, 77, 66 Maths: 65, 66, 58, 60, 61, 70, 74, 66, 69, 68, 71, 55, 51, 49, 50, 71, 99, 85, 70, 66 a Display the data in a back-to-back stem-and-leaf plot. b For each subject find the median. c For each subject state the range. d For each subject find the interquartile range.

4C

11 Betty runs a surf and ski shop. The table below shows the monthly sales of both types of equipment.

4C

Month

Surf sales ($)

Ski sales ($)

January

20 000

5 000

February

18 000

6 000

March

12 000

8 000

April

9 000

10 000

May

6 000

12 000

June

4 000

12 000

July

5 000

9 000

August

8 000

8 000

September

10 000

6 000

October

11 000

3 000

November

15 000

4 000

December

22 000

9 000

a Display both sets of data on the same radar chart. b Use the chart to compare trends in the sales. 12 The data below give the cost per minute of a long-distance telephone call with three companies. Telecomm

Omtus

Tel One

Day

21c

25c

17.5c

Economy

18c

15c

17.5c

Night

12c

12c

17.5c

Display this information in an area chart.

4C

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4D

13 The stem-and-leaf plot below compares the crowds (correct to the nearest thousand) at a football team’s home and away matches. Key: 2 5 = 25 000 Home Away 8 0 67 732 1 0116899 6632 2 45 552 3 a Calculate the median of both data sets. b Calculate the range of both data sets. c Calculate the interquartile range of both data sets. d Display both sets of data on a box-and-whisker plot.

4D

14 The figure on the right shows a 2003 box-and-whisker plot showing 2004 the average number of weekly car sales made in 2003 and 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Scale 2004. a What was the median for each year? b In which year was the range of sales greatest? c In which year was the interquartile range of sales greatest? d In which year did the car yard perform better? Explain your answer.

4D

15 The two-way table below compares the number of men and women who are right- and lefthanded.

Right-handed Left-handed Totals

Men

Women

Totals

158

172

330

17

15

32

175

187

362

a What percentage of males are left-handed? b What percentage of females are left-handed? c Based on the above data, is there any significant difference between the percentage of male and female left-handers?

4D

16 Hsiang compares her marks in 10 English exams and 10 Maths exams. English: 76, 74, 80, 77, 73, 70, 75, 37, 72, 76 Maths: 80, 56, 92, 84, 65, 58, 55, 62, 70, 71 a Calculate Hsiang’s mean mark in each subject. b Calculate the range of marks in each subject. c Calculate the standard deviation of marks in each subject. d Based on the above data, in which subject would you say that Hsiang performs more consistently?

Chapter 4 Interpreting sets of data

165

Practice examination questions 1 multiple choice The table below shows the number of patients seen each day by a local doctor. No. of patients

Frequency

12

3

13

8

14

15

15

23

16

18

17

13

Which of the following statements are correct? A The range of the data is 20. B The mean of the data is 15.05. C The standard deviation of the data is 1.34. D The median of the data is 15. 2 multiple choice The data below show the number of people that live in each house in a small street. 4, 4, 5, 3, 2, 5, 11, 2 The outlier in this data set has: A the greatest effect on the mean. B the greatest effect on the median. C the greatest effect on the mode. D an equal effect on the mean, median and mode. 3 multiple choice

Frequency

The two data sets below show the number of goals scored in 15 matches by two soccer teams. Manchester: 0, 2, 1, 2, 1, 6, 0, 0, 1, 5, 0, 0, 1, 1, 1 Liverpool: 5 4 3 2 1 0

0 1 2 3 4 5 Number of goals

Which of the following statements is correct? A The Manchester data are negatively skewed, while the Liverpool data are positively skewed. B The Liverpool data are negatively skewed, while the Manchester data are positively skewed. C Both sets of data are positively skewed. D Both sets of data are negatively skewed.

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4 The two-way table below shows the number of men and women who work in excess of 45 hours per week. Men

Women

Totals

£ 45 hours

132

128

260

> 45 hours

69

34

103

Totals

201

162

363

The percentage of men who work greater than 45 hours per week is closest to: A 28% B 34% C 51% D 67% 5 multiple choice The figures below show the number of attempts that the boys and girls in a Year 12 class take to get their driver’s licence. Boys: 1, 2, 4, 1, 1, 2, 1, 1, 2, 2, 3, 1 Girls: 2, 2, 4, 2, 1, 2, 2, 3, 1, 1, 1, 2 When comparing the performance of the boys and the girls, it is found that the boys have: A a lower mean and a lower standard deviation B a lower mean and a higher standard deviation C a higher mean and a lower standard deviation D a higher mean and a higher standard deviation 6 The data below show the weekly income among ten Year 12 boys and girls. Boys: $80, $110, $75, $130, $90, $125, $100, $95, $115, $150 Girls: $50, $80, $75, $90, $90, $60, $250, $80, $100, $95 a Calculate the median of both sets of data. b Calculate the range of both sets of data. c Calculate the interquartile range of both sets of data. d Display both sets of data on a box-and-whisker plot. e Use the statistics function on the calculator to find the mean and standard deviation of both sets of data. f Discuss whether the boys or girls have a more consistent average weekly income.

CHAPTER

test yourself

4

7 In the week leading up to the NRL grand final, Kylie records the number of points scored by both teams in each game throughout the season and displays the information on the stem-andleaf plot below. Key: 1 8 = 18 Sharks Bulldogs 8 0 84422 1 5558889 88644432200 2 0022226668889 886200 3 000222 862 4 a Find the median of both sets of data. b Which team’s scores are the more consistent? c Describe the skewness of the Sharks’ scores. d Find the mean and standard deviation of the Bulldogs’ scores.

Algebraic skills and techniques

5 syllabus reference Algebraic modelling 3 • Algebraic skills and techniques

In this chapter 5A 5B 5C 5D 5E

Substitution Algebraic manipulation Equations and formulas Solution by substitution Scientific notation

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

5.1

Substitution into a formula

1 For each of the following formulas, find the value of the subject, given the variables. a I = Prn given that P = 5750, r = 0.08 and n = 5 (answer correct to 1 decimal place) b A = ab given that a = 5.6 and b = 9.2 h c A = --- (df + 4dm + dl) given that h = 60, df = 0, dm = 32 and dl = 28 3 d A = 1--- ab sin C given that a = 23.4, b = 37.1 and C = 60˚ (answer correct to 1 decimal place) 2

5.2

Simplifying like terms

2 Simplify each of the following expressions. a r+r+r+r+r b 7m + 9m – 6m d 9a + 2b – 8a – 7b e 2x + 4y – 3x

5.3

Multiplication using indices

3 Simplify each of the following expressions. a r4 r6 b 6a3 3 2 4 d 5q 7q e 12m 4m5

5.4

c 9x + 7 + 8 + 7x f 5m – 5n + 4m – 3n

c 4p5 7p f 3r2s5 9rs6

Division using indices

4 Simplify each of the following expressions. 6

8

4q d -------32q

5.5

4

42x c ----------x

6

4k f --------32k

30m b ------------5

d a ----2d

56rs e ------------7rs

c (2c2)4

Solving linear equations

6 Solve each of the following equations. a z – 42 = 76 b 4y = 96 9v d ------ = 8 e 6(t – 5) = 54 6

5.7

3

Raising a power to a power

5 Simplify each of the following expressions. a (a2)4 b (4b)3

5.6

6

c 6w – 9 = 69 f 20 + 2n = n + 54

Changing the subject of a formula

7 Write each of the following in scientific notation. a 25 000 b 236 000 000 c 400 000

d 26 000 000 000 000

Chapter 5 Algebraic skills and techniques

169

Substitution During the preliminary course we studied substitution. Substitution involves replacing a pronumeral in an expression with a numerical value. There are many different types of expressions that may need substitution. A linear expression such as 3x + 5 involves no index other than 1. When graphed, these expressions form a straight line. When performing a substitution, we write the expression and the values of the known pronumerals, rewrite the expression having substituted the given values, and finally calculate the value of the expression.

WORKED Example 1

h The formula for the area of a trapezium is given by A = --- ( a + b ) , where a and b are the 2 parallel sides and h is the height. Find the area of a trapezium with parallel sides 4.2 cm and 7.9 cm and a height of 5.1 cm. THINK

WRITE

1

Write down the given expression.

2

Write down the variables where the values are known. Substitute the given values into the formula. Evaluate.

3 4

h A = --- ( a + b ) 2 a = 4.2, b = 7.9, h = 5.1 5.1 A = ------- ( 4.2 + 7.9 ) 2 A = 30.855

the graphics calculator Graphics Calculator tip! Using to substitute You can use a graphics calculator to assist with substitutions. Consider worked example 1. 1. From the MENU select RUN.

2. Assign the value a = 4.2 by entering 4.2 ALPHA [A] and then pressing EXE .

3. Repeat step 2 to assign the values b = 7.9 and h = 5.1.

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Maths Quest General Mathematics HSC Course

4. Now enter the expression H ÷ 2 press EXE .

(A + B) and then

Many expressions involve higher powers or indices. An expression involving a power of 2 is called a quadratic expression, an expression involving a power of 3 is called a cubic expression. Consider the expression M = 6r2. In this expression, only the pronumeral r is raised to the power of 2. In the expression M = (6r)2 the product 6 r is raised to the power of 2.

WORKED Example 2 The expression V = 4--3- p r 3 is used to calculate the volume of a sphere. Find the volume of the sphere with a radius of 4.2 cm, giving your answer correct to 1 decimal place. THINK

WRITE 4 --3

r3

1

Write down the given expression.

V=

2

Write down the variables where the values are known. Substitute the given values into the formula. Evaluate.

r = 4.2

3 4

V=

4 --3

( 4.2 ) 3

V = 310.3 cm3

Other expressions may involve taking square roots and cube roots. Care must be taken to use the calculator correctly. The square or cube root must be taken of the entire part of the expression that is under the root sign. This may involve using brackets.

WORKED Example 3 3V ------- is used to calculate the radius of a sphere, given the volume. 4 Find the radius of a sphere with a volume of 200 cm3. (Give your answer correct to 1 decimal place.) The expression r =

3

THINK 1

Write down the given expression.

2

Write down the variables where the values are known.

3

Substitute the given values into the formula. Evaluate.

4

WRITE 3V ------4 V = 200 r=

3

r=

3

3 200 -----------------4

r = 3.6 cm

Chapter 5 Algebraic skills and techniques

171

remember 1. Substitution involves replacing a pronumeral or pronumerals in an expression with numerical values. 2. Linear expressions involve only powers of 1. 3. Quadratic and cubic expressions involve powers of 2 and 3 respectively. In these expressions be sure to raise only the relevant part of the expression to the power. 4. Expressions that involve square and cube roots must be solved by correctly using a calculator and brackets.

5A Example

5.1

2 Find the value of each of the following by substituting into the formula. a A = 1--- bh, if b = 5 and h = 12.3 2

Substitution into a formula L Spre XCE ad

Substitution

3 The formula P = 2l + 2w is used to find the perimeter of a rectangle. Use the formula to find the perimeter of a rectangle, where l = 3.5 and w = 9.7. 4 The formula C = p d is used to calculate the circumference of a circle. Use the formula to find the circumference of a circle with a diameter of 9.5 m. Give your answer correct to 1 decimal place. WORKED

Example

2

5 In the formula A = 6s2, find the value of A when s = 5.5. 6 Find the value of each of the following by substituting into the formula. (Give your answers correct to 2 decimal places.) a V = r2h, if r = 0.75 and h = 2.5 b A = p (R2 - r2), if R = 2.2 and r = 1 c V = 4--- p r 3, if r = 3.2 3 2

d P = I R, where I = 0.6 and R = 230 e E = 1--- mv2, where m = 23 and v = 4.7 2

7 The formula A = 2p r 2 + 2p rh is used to calculate the surface area of a cylinder. Calculate the surface area of a cylinder with a radius of 2.3 cm and a height of 6.4 cm. Give your answer correct to the nearest whole number.

WORKED

Example

3

8 Use the formula S = ut + 1--- at2 to calculate the value of S, when u = 9, t = 5 and 2 a = 4.5. A 9 Use the formula r = ------ to find the value of r (correct to 1 decimal place) when 4 p A = 500.

sheet

PRT b A = P + ----------- , if P = 2000, R = 6.55 and T = 2.5 100 c S = 2(lw + lh + wh), where l = 3, w = 2.5 and h = 1.1 d V = u + at, where u = 20, a = 4 and t = 25 e T = a + (n - 1)d, if a = 66, n = 56 and d = -4

SkillS HEET

1

1 In the formula A = l ¥ b, find the value of A, given that l = 6.7 and b = 4.8.

E

WORKED

Substitution

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Maths Quest General Mathematics HSC Course

10 Find the value of each of the following by substitution into the formula. Where necessary, give your answer correct to 1 decimal place. a c =

a 2 – b 2 , when a = 17 and b = 8

b T = 2

3V ------- , if V = 600 and h = 25 h

c S = d r = e m =

L --- , when L = 65 and g = 9.8 g

3

3V ------- , if V = 900 4 2xy + y 3 , when x = 2 and y = 3

A --- is used to calculate the side 6 length of a cube having been given the area. Calculate the side length of a cube with a surface area of 162.24 cm2.

11 The formula s =

12 The formula A = s ( s – a ) ( s – b ) ( s – c ) can be used to find the area of any triangle, where a, b and c are the side lengths and s is half the perimeter of the triangle. Given that the side lengths of a triangle are 4 cm, 8 cm and 9 cm: a+b+c a calculate the value of s s = --------------------2 b find the area of the triangle, correct to 1 decimal place.

Algebraic manipulation Basic manipulation of algebraic expressions was covered in the preliminary course. We need to be able to add and subtract algebraic expressions as well as multiply and divide them. Algebraic expressions are added and subtracted by collecting like terms. Only the same pronumeral or combination of pronumerals can be added together.

WORKED Example 4

Simplify each of the following. a 8x + 2x 11x b 9a2 + 2a + 4a2 7a THINK

WRITE

a Each term uses the same pronumeral and so we add and subtract the coefficients.

a 8x + 2x

b

b 9a2 + 2a + 4a2 7a = (9a2 + 4a2) + (2a 7a) = 13a2 5a

1

2

Rewrite the expression, grouping the like terms together. Remember that a2 and a are not like terms. Complete each addition and subtraction separately.

11x = x

Chapter 5 Algebraic skills and techniques

173

To multiply and divide algebraic expressions, we need to remember the index laws covered in the preliminary course. First Index Law:

ax

ay = ax + y

Second Index Law: a x ÷ a y = a x Third Index Law:

y

ax or -----y = a x – y a

(a x) y = a xy

When multiplying and dividing algebraic expressions it is important to remember to apply the index laws separately to each pronumeral.

WORKED Example 5 Simplify each of the following fully. 48 p 2 q 4 a 6m7 7m3 b ----------------c (5x4)3 6 pq 3 THINK

WRITE

a Multiply the coefficients and add the indices.

a 6m7

b Divide the coefficients and subtract the indices for each pronumeral separately.

48 p 2 q 4 b ------------------ = 8 pq 6 pq 3

c Calculate 53 and multiply the indices.

c (5x4)3 = 125x12

7m3 = 42m10

The manipulation of algebraic expressions will also involve the expansion of brackets. When expanding brackets, we multiply every term inside the brackets by the term immediately outside the brackets.

WORKED Example 6 Expand 2x3(6xy

9y4).

THINK

WRITE

Multiply both terms inside the brackets by 2x3.

2x3(6xy

9y4) = 12x4y

18x3y4

remember 1. Algebraic expressions are added and subtracted by collecting like terms. 2. Algebraic expressions are multiplied and divided using the index laws. • First Index Law: ax ay = ax + y ax • Second Index Law: a x ÷ a y = a x y or -----y = a x – y a • Third Index Law: (a x) y = a xy 3. When using the index laws, apply each law to each pronumeral separately. 4. When expanding brackets, multiply every term inside the brackets by what is immediately outside.

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Maths Quest General Mathematics HSC Course

5B SkillS

HEET

5.2

1 Simplify each of the following expressions. a 4a + 8a b 14b + 4b 4a d 35d + 6d e 5e - e g g - 8g h -4h + 9h j 7j - 5j + 9j k -3k + 8k - k

WORKED

Example

Simplifying like terms

SkillS

HEET

5.3

et

EXCE

c f i l

23c - 9c 16f - 15f -2i - 7i 5l - 15l + 8l

2 Simplify each of the following expressions. a 8m + 3m - 9 b n - 4 + 7n 4b d 5r + 9s - 2r + 2s e 7t + 1 - 4t - 7 g 4w2 - 3w3 + 2w2 - w3 h 2xy + 4xz - 3xy + xz j 4x + 3y - 2xy + 6x - 4yx + y

c 7p2 + 6p + 3p2 - 2p f 6u - 8v - 7u + 2v i 4p2 - 12 + p2 - 4

3 Simplify each of the following. a a5 ¥ a8 b b ¥ b3 5a d d3 ¥ 7d e 4p4q3 ¥ 3p5q2 4 5 g 4mn ¥ 7m n h 4p5 ¥ 5q4 3 3 2 4 5 j 6u v ¥ 4v w ¥ 2uw

c 3c2 ¥ 4c5 f 7gh ¥ 9g2h3 i 6xyz ¥ 4x2y2

WORKED

Example

Multiplication using indices

reads L Sp he

Algebraic manipulation

WORKED

Example

Index laws

4 Simplify each of the following. a k4 ÷ k b 15m7 ÷ 5m2 5b 14x 5 56m 4 n 3 d ----------e -----------------7x 7m 2 n g m6n7 ÷ mn h 48p3 ÷ 6q3 3 4 6 2 2 2 j 32p q r ÷ 4pqr ÷ 2p q

WORKED

SkillS

Example

HEET

5.4

SkillS

Division using indices HEET

5.5

5 Simplify each of the following. a (a3)4 b (2b4)2 5c 2 3 2 d (4x y ) e (2pq2)4

Work

5.1

45x 5 ----------9 i 121a ÷ 11b f

WORKED

Example

6 Expand each of the following. a 2(m + 5) b x(x + 2) 6 d 3q2(6q4 - 2) e 5n(m - 5n) g -3(d + 5) h -3m(m - 2n) j 6pqr(3pq - r)

c (3m2)3

WORKED

Example

Raising a power to a power

T SHEE

c 48n7 ÷ 8n2

c 3a(3a + 2b) f 7a2b4(2a4 - 3b6) i -6r3(2 - 3r3)

7 Expand and simplify each of the following. a 4(x + 2) + 2(3x - 1) b a(a + 7) + 2(3a - 5) d 5(4x - 7) - 2(x + 5) e 2p(2p - 5) - 5(p - 6)

c 2m(m - n) + 6n(m - 2n) f 2xy(3x - 4y) - x(y - xy)

8 Fully simplify each of the following. a a 4 ¥ a 5 ÷ a2 b (m2)3 ÷ m4 ¥ m

c 32m2n3 ÷ 4mn ¥ 2n3

d 4x6y7 ¥ 5xy4 ÷ 2x6

e (2z3)4 ÷ 8z5 ÷ 2z7

f

9m 2 n 4 ¥ 4mn 2 ----------------------------------( 6mn 3 ) 2

Chapter 5 Algebraic skills and techniques

175

1 a 1 Calculate the value of S = ----------- , when a = 8 and r = 0.2. 1–r 2 Calculate the value of S = ut + --1- at2, when u = 4.5, t = 6.1 and a = 4. 2

3 Calculate the value of S =

3V ------- , when V = 352.6 and h = 4.5. (Give your answer h

correct to 1 decimal place.) Simplify the following expressions. 4 6x

7x + x

5 4a + 2b 6 3b4

3a

5x 8b

5b2

42g 3 h 4 7 ---------------7h 2 8 (5p3q4)2 9 5x(2 10 3a(2a

x) 5b)

4b(a

6b)

Equations and formulas During the preliminary course we studied equations. The object of solving an equation was to find the value of an unknown pronumeral that made that statement true. In solving the equation we reversed every process that had been performed on the pronumeral until it became the subject of the equation. In many cases, an equation arose as the result of substitution into a formula.

WORKED Example 7

In the formula C = 2 r, find the value of r when C = 100, correct to 2 significant figures. THINK 1 2 3

Write the formula. Substitute the value of C. Divide each side by 2 and round the answer off to 2 significant figures.

WRITE C=2 r 100 = 2 r 100 r = --------2 = 16

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Maths Quest General Mathematics HSC Course

equations that arise Graphics Calculator tip! Solving from substitution You can use the equation solver function on a graphics calculator to solve equations that arise as the result of a substitution. Consider worked example 7. 1. From the MENU select EQUA.

2. Press F3 (Solver).

3. Delete any existing equation, and enter the equation that arises after the substitution is made. To enter 100 = 2 r press 1 0 0 SHIFT [=] 2 SHIFT [ ] ALPHA [R]. Note: At this stage you may have a different value of R, but this is to be ignored. 4. Press F6 to solve the equation.

Some equations involve powers and roots. In the solution to an equation, remember that the opposite function to taking a square is to take the square root and vice versa. When solving such an equation, both the positive and negative square roots are possible solutions. For example, the equation x2 = 9 has the solution x = ±3. This differs from 9 , which equals 3. The square root symbol indicates to take the positive square root only.

WORKED Example 8 In the equation d = 5t2, find the value of t when d = 320. THINK 1 2 3 4

Write the formula. Substitute the value of d. Divide each side by 5. Take the square roots of each side, considering both the positive and negative answers.

WRITE d = 5t2 320 = 5t2 t2 = 64 t = ±8

Chapter 5 Algebraic skills and techniques

177

Note: If you use the solver function on your graphics calculator, only the positive solution is given. It is important that you remain aware that equations of this type have a positive and negative solution. With such examples, we consider both the positive and negative cases only where appropriate. In practical cases where measurements are being considered, only the positive answer is given. Using the same process as this we can change the subject of a formula. The subject of the formula is the single pronumeral usually written on the left-hand side of the formula. For example, in the formula A = r 2, A is the subject. We are able to make another pronumeral the subject of the equation by moving all other numbers and pronumerals to the other side of the formula, as if we were solving an equation. Formulas that need the subject changed include those with both linear and quadratic terms.

WORKED Example 9 Make x the subject of the formula y = 5x THINK

2. WRITE

1

Write the equation.

y = 5x

2

Add 2 to each side.

y + 2 = 5x

3

Divide each side by 5 (and write the new subject of the formula on the left-hand side).

2

y+2 x = -----------5

This method is also used for quadratic formulas but, as with equation solving, we must remember to use both the positive and negative square root where appropriate.

WORKED Example 10 The formula A = 4 r 2 is used to find the surface area of a sphere. Make r the subject of the formula. THINK 1

Write the formula.

2

Divide both sides by 4 .

3

Take the square root of each side. As r is the radius, a length, we consider only the positive square root.

WRITE A = 4 r2 A ------ = r 2 4 r=

A -----4

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Maths Quest General Mathematics HSC Course

remember 1. An equation can be formed after substitution into a formula. 2. When solving an equation, the object is to find the value of the unknown. 3. When an equation involves taking a square, the opposite function used to solve the equation is a square root. 4. Both the positive and negative square root should be taken unless the context of the equation means that only the positive should be used. 5. To make another pronumeral the subject of an equation, the same methods as for equation solving are used although we use pronumerals rather than make actual calculations.

SkillS

5C HEET

5.6

WORKED

Example

7

SkillS

Solving linear equations HEET

Equations and formulas

1 The formula C = d is used to calculate the circumference of a circle. Find the diameter of a circle that has a circumference of 40 cm. Give your answer correct to 3 significant figures. 2 The formula P = 2l + 2w is used to calculate the perimeter of a rectangle. Calculate the length of a rectangle that has a perimeter of 152 m and a width of 38 m.

5.7

3 In each of the following, find the value of the unknown after substitution into the formula. Where appropriate, give your answer correct to 1 decimal place. h a A = --- ( a + b ) ; find h when A = 145, a = 15 and b = 25. 2 b A = l w; find w when A = 186 and l = 15. c V = r 2h; find h when V = 165.2 and r = 3.6. d T = a + (n 1)d; find n when T = 260, a = 15 and d = 11. e v2 = u2 + as; find s when v = 5.5, u = 2.4 and a = 1.2.

Changing the subject of a formula

WORKED

Example

8

4 In the formula A = 6s2, find the value(s) of s when A = 150. 5 The formula A = r 2 is used to calculate the area of a circle. Find the radius of a circle, correct to 2 decimal places, given that the area of the circle is 328 cm2. 6 Substitute into each of the formulas and solve the equation to find the value of the unknown. Where necessary, give your answer correct to 2 decimal places. a V = r2 h; find r when V = 1.406 25 and h = 2.5. b A = (R2 r2); find R when A = 12 and r = 1. c V=

4 --3

r 3, find r when V = 136.

d E = --1- mv2; find v when E = 254 and m = 23. 2

e P = I 2R; find I when P = 0.54 and R = 1.5. WORKED

Example

9

7 Make x the subject of the formula y = 2x + 1. 8 Make l the subject of the formula A = l

b.

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Chapter 5 Algebraic skills and techniques

h 9 In the formula A = --- ( a + b ) : 2 a make a the subject of the formula b make h the subject of the formula. WORKED

Example

10 Make r the subject of the formula A = r 2.

10

11 In the formula E = mc2: a make m the subject of the formula b make c the subject of the formula. Questions 12 to 14 refer to the following information. The volume of a square-based pyramid with the side of the base, s, and the height, h, is given by the formula V = 1--- s2h. 3

12 multiple choice The side length of the base of a square-based pyramid with the height, h, and volume, V, is given by: V A s = 3 ---h

B s =

h ------3V

V C s = -----3h

D s =

3V ------h

13 multiple choice The height of a square-based pyramid with the side of the base 5 cm and the volume 75 cm3 is: A 8 cm B 9 cm C 10 cm D 12 cm 14 multiple choice If both the side of the base and the height are doubled the volume is: A doubled B increased by 4 times C increased by 6 times D increased by 8 times 15 In each of the following, make the subject of the formula the pronumeral indicated in brackets. a V = r 2 [r] d T = 2

L --g

[L]

b v2 = u2 + as

[u]

e c2 = a2 + b2

[a]

c V=

4 --3

r 3 [r]

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Solution by substitution From our earlier work on equations, we have found that substituting the solution back into the original equation can check the answer to the equation. If the solution is correct, then the value that is substituted will satisfy the equation. For example, consider the following equation. 4x 5 = 19 4x = 24 x=6 Substituting x = 6 into 4x + 5 = 19 Left-hand side (LHS) = 4 6 5 Left-hand side (LHS) = 19 Left-hand side (LHS) = Right-hand side (RHS) By substitution we can see that x = 6 is the correct solution to this equation. Some more difficult equations can have an approximate solution found by substituting a first guess into the equation and gradually refining the solution.

WORKED Example 11

Find an approximate solution to the equation 2x = 20 (correct to 1 decimal place). THINK 1 2 3 4 5 6

WRITE

Make a first guess (x = 4) and substitute into the equation. As 24 < 20, make a second guess that is greater than 4 (x = 5). As 25 > 20, make the next estimate between 4 and 5 (x = 4.5). As 24.5 > 20, make the next estimate between 4 and 4.5 (x = 4.3). As 24.3 < 20, make the next estimate between 4.3 and 4.5 (x = 4.4). Since x = 4.3 gives a result closer to 20 than x = 4.4, the solution, correct to 1 decimal place, is x = 4.3.

Test x = 4 24 = 16 Test x = 5 25 = 32 Test x = 4.5 24.5 = 22.6 Test x = 4.3 24.3 = 19.7 Test x = 4.4 24.5 = 21.1 Solution is x = 4.3.

Many such equations will arise from a practical situation such as investments.

WORKED Example 12

Terry has $1000 to invest; however, he needs $1500 to purchase the electric guitar that he wants. If Terry invests his $1000 at 6% p.a., the amount in the account at any time can be found using the formula A = 1000(1.06)n, where n is the number of years for which the money has been invested. Find how long it will take (correct to the nearest year) for Terry’s $1000 to grow to $1500.

Chapter 5 Algebraic skills and techniques

THINK 1 2 3 4 5 6

7

181

WRITE

Write the formula. Substitute A = 1500. Divide both sides by 1000. Make a first estimate for the solution (n = 5). As (1.06)5 < 1.5, make a second estimate greater than n = 5 (n = 8). As (1.06)8 > 1.5, make the next estimate between n = 5 and n = 8 (n = 7). The solution must be n = 7 as (1.06)7 = 1.5, correct to 2 decimal places.

A = 1000(1.06)n 1500 = 1000(1.06)n 1.5 = (1.06)n (1.06)5 = 1.34 (1.06)8 = 1.59 (1.06)7 = 1.50

It will take 7 years for the $1000 to grow to $1500 at 6% p.a

This type of question can be solved using a graphics calculator and by setting up a table of values.

Graphics Calculator tip! Using the table function We can use the table function on a graphics calculator to take the repetition out of solution by substitution. Consider worked example 12 above. 1. From the MENU select TABLE.

2. For Y1 enter the function 1000 EXE .

1.06^X and press

3. Press F5 (RANG) and then enter the settings shown on the screen opposite. These determine the start and end values and the pitch of the table. The pitch is the increment by which x changes from the start value. 4. Press EXE to return to the previous screen, and then press F6 (TABL). You will then need to scroll down to find the value of Y1 that is closest to 1500.

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remember

1. Equations such as 2x = 10 have no opposite operation that you can use easily. Find an approximate solution by substitution and then improve on the first estimate. 2. The first estimate is taken and substituted into the equation. A second estimate, either higher or lower than the first depending on the result of the substitution, is then taken. 3. Further estimates can then be taken by dividing the range within which we know the solution lies. 4. Most equations of this type can be solved using the table function on a graphics calculator.

5D WORKED

Example

11

Solution by substitution

1 Solve the equation 2x = 100, correct to 1 decimal place. 2 Solve the equation 1.1x = 2, correct to the nearest whole number. 3 Solve the equation 0.9x = 0.5, correct to the nearest whole number.

4 The amount of time that it will take for an investment to double when invested at 5% p.a. can be calculated using the equation (1.05)n = 2. Find the value of n, correct to 12 the nearest whole number.

WORKED

Example

5 It is anticipated that the value of a house will keep pace with inflation. Judy purchased a house in 2001 for $265 000. The future value of the house can be calculated using the r n formula A = P 1 + --------- , where A is the future value, P is the present value, r is the 100 inflation rate and n is the number of years. Judy wants to know how many years it will take for the value of her property to exceed $500 000 given that the inflation rate will average 4% p.a. a Substitute the known values into the formula to create an equation. b Solve the equation for n, correct to the nearest whole number. 6 The value of a computer decreases at the rate of 30% p.a. A new computer purchased for $3000 can have its value after n years calculated using the formula V = 3000(0.7)n, where n is the age of the computer in years. Calculate when the value of the computer will equal $500, correct to the nearest year. 7 The distance through which an object will fall in t seconds can be calculated using the formula d = 5t2. a Copy and complete the table below. t

1

2

3

4

5

6

7

8

9

10

d b Calculate the length of time that it will take an object to fall 300 m, correct to the nearest second.

Chapter 5 Algebraic skills and techniques

183

8 Kayla has 80 m of fencing in which to enclose a rectangular area for a garden. a Copy and complete the table below. Length

5

10

15

20

25

30

35

Width Area b What dimensions should the garden be if it is to enclose the maximum possible area? c The garden is to use an existing fence for one side and use the 80 m of fencing to build the other three sides. Draw up a table to calculate the dimensions Kayla should now build the garden to maximise the area.

Repeated enlargements Consider the following problem. I need to enlarge a diagram on my photocopier to twice its original size. My photocopier can enlarge to only 150% of the original. Explain how I can make the enlargement that I need by using a repeated enlargement.

2 1 Given that y = 25

5x, find the value of y when x = 3.

2 Given that T = 6n + 5n2, find the value of T when n = 2. 3 In the formula y =

r 2 – x 2 , find y when r = 5 and x = 4.

4 Given that C = d find d, correct to 1 decimal place, when C = 400. 5 Given that d = 5t 2, find t when d = 2000. Simplify the following expressions. 6 4a + 6b 7 4x2y3

3a

9b

5x5y

8 (4m3n4)3 18m 2 n 4 9 -----------------6mn 2 10 Use the method of substitution to solve the equation (1.1)x = 3, correct to the nearest whole number.

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Scientific notation Scientific notation is used to express very large or very small numbers in terms of a power of 10. It is particularly useful in branches of science such as astronomy, where large distances are measured, or in biology, where very small measurements of microbes are taken. As we found in the preliminary course, numbers are written in scientific notation by rewriting the number with a decimal point after the first significant figure. This decimal is then multiplied by the appropriate power of 10. This power of 10 is found by counting the number of places that the decimal point has been moved. When moving the decimal point left, the power of 10 is positive; it is negative when moving the decimal point to the right.

WORKED Example 13 Write each of the following in scientific notation. a 8 000 000 b 13 400 000 000

c 0.000 034 51

THINK

WRITE

a

Move the decimal point after the first significant figure. The decimal point has been moved 6 places left.

a 8 000 000 = 8

Move the decimal point after the first significant figure. The decimal point has been moved 10 places left.

b 13 400 000 000 = 1.34

Move the decimal point after the first significant figure. The decimal point has been moved 5 places right.

c 0.000 034 51 = 3.451

1

2

b

1

2

c

1

2

106

1010

10

5

In many examples we are required to round such measurements off to a given number of significant figures.

WORKED Example 14 Write each of the following measurements in scientific notation, correct to 3 significant figures. a 97 856 472 124 km b 0.000 000 124 117 23 mg THINK

WRITE

a

a 9.79

1 2 3

Move the decimal point after the first significant figure. The decimal point has been moved 10 places left. Round the decimal off after the third significant figure.

1010 km

Chapter 5 Algebraic skills and techniques

THINK

WRITE

b

b 1.24

1

Move the decimal point after the first significant figure.

2

The decimal point has been moved 7 places left.

3

Round the decimal off after the third significant figure.

10

7

185

mg

To change a number from scientific notation back to a decimal number, move the decimal point to the right if the power of 10 is positive. If the power of 10 is negative, move the decimal point to the left. Zeros will need to be added if there are insufficient decimal places.

WORKED Example 15 Write each of the following as a decimal number: a 3.85 108 b 8.654 106 THINK

WRITE

a Move the decimal point eight places to the right. You will need to add six zeros to do this.

a 3.85

b Move the decimal point six places to the left. You will need to add five zeros after the decimal point to do this.

b 8.654

108 = 385 000 000

10–6 = 0.000 008 654

remember 1. Scientific notation is used as a shorthand way of writing very large and very small numbers. 2. The decimal point is placed after the first significant figure, and then this decimal is multiplied by the appropriate power of 10. 3. The power of 10 is found as follows. • If the decimal point is moved left (for large numbers), the power of 10 is the number of places moved left. • If the decimal point is moved right (for small numbers), the power of 10 is negative and is the number of places moved right. 4. To write a number given in scientific notation as a decimal number, move the decimal point: (a) to the right for a positive power of 10 (b) to the left for a negative power of 10.

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5E SkillS

HEET

1 Write each of the following in scientific notation. a 90 000 b 20 000 000 000 13a Scientific WORKED 2 Write each of the following in scientific notation. notation Example a 1 458 000 b 23 650 000 000 000 13b

5.8

WORKED

Example

c 700 c 2589

3 Write each of the following in scientific notation. a 0.000 000 02 b 0.004 57 c 0.000 000 000 049 321 13c Scientific 4 Write each of the following in scientific notation, correct to 3 significant figures. notation WORKED Example a 93 154 789 km b 78 548 963 214 mm c 45 874 t 14 d 0.003 654 7 g e 0.213 658 mL f 0.000 005 687 4 s et

reads L Sp he

EXCE

Scientific notation

WORKED

Example

5 Write each of the following as a decimal number. a 3.4 ¥ 104 b 2.87 ¥ 106 15a WORKED 6 Write each of the following as a decimal number. Example a 5.85 ¥ 10–4 b 1.97 ¥ 10–6 15b WORKED

Example

c 3.0248 ¥ 1010 c 1.002 ¥ 10–3

7 An astronomical unit (AU) is defined to be the distance between the Earth and the sun and is equal to approximately 150 000 000 km. The table below shows the distance between each planet in the solar system and the sun in astronomical units. Write the distance between each planet and the sun in kilometres in scientific notation, correct to 3 significant figures.

Work

Planet

T SHEE

5.2

Distance (AU)

Mercury

0.39

Venus

0.72

Earth

1.0

Mars

1.52

Jupiter

5.20

Saturn

9.54

Uranus

19.18

Neptune

30.06

Distance in km (scientific notation)

8 Complete each of the measurement conversions. a 2.35 ¥ 107 mm = ___ m b 8.4 ¥ 107 m = ___ km 5 c 6.4 ¥ 10 cm = ___ mm d 6.58 ¥ 106 kg = ___ t 6 e 7.802 ¥ 10 t = ___ kg f 8.29 ¥ 1010 kg = ___ g 8 g 1.87 ¥ 10 L = ___ kL h 2.178 ¥ 107 kL = ___ L i 5.55 ¥ 107 L = ___ mL

Chapter 5 Algebraic skills and techniques

187

summary Substitution • Substitution involves the replacement of a pronumeral with a numerical value in an expression. • These expressions include linear expressions that have only powers of 1, quadratics that have a power of 2 and cubics that have a power of 3. • Care must be taken when using a calculator to apply the power to the correct term.

Algebraic manipulation • Algebraic terms are added or subtracted by collecting like terms. • Algebraic terms are multiplied or divided by applying the index laws to each pronumeral separately. First Index Law: ax ay = ax + y Second Index Law: ax ÷ ay = ax y Third Index Law: (a x ) y = a xy

Equations and formulas • After substituting into a formula, an equation will be created when you are not finding the subject of the formula. • The equation that you may need to solve could be linear or quadratic. • Using the same method as for solving equations, you can rearrange a formula to make another pronumeral the subject of the formula.

Solution by substitution • Some equations have no opposite operation that allows you to easily solve the equation. These equations can have an approximate solution found using substitution. • To solve an equation using this method, make a first estimate of the solution and substitute that estimate into the equation. • Use the result of that substitution to make an improved estimate, and then substitute the improved estimate into the equation. Repeat this process until a solution to the desired degree of accuracy is found.

Scientific notation • Scientific notation is used to write very large or very small numbers in a shorthand way using powers of 10. • The decimal point is moved after the first significant figure and is multiplied by the appropriate power of 10. • For large numbers, the power of 10 is the number of places the decimal point has been moved to the left. • For small numbers, the power of 10 is negative and is the number of places the decimal point has been moved to the right.

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CHAPTER review 5A 5A 5A 5A

1 Find the value of V = A

5B

5 Simplify each of the following by collecting like terms. a m+m+m+m+m b 7q + 9q d 23t 22t e 4m + 6n 2n g 11k 6l + 4l 8k h 5x2 + 20x + 3x2 6x

5B

6 Simplify each of the following. a 4a4 7a5 b 5b 9b d 12m5n6 mn e 42x6 ÷ 7x4 28q 2 g ----------h (3p2q4)3 4q

h, when A = 54 and h = 3.

2 Find the value of S = ut + --1- at2, when u = 4.1, t = 6.2 and a = 0.6. 2

3 Find the value of d =

( x 1 – x 2 ) 2 + ( y 1 – y 2 ) 2 , when x1 = 2, y1 = 7, x2 = 3, and y2 = 5.

4 Find the value of each of the following giving your answer, where necessary, correct to 2 decimal places. a A = r(r + s), when r = 3.9 and s = 7.2 b C = 5--- (F 32), when F = 100 9 a c S = ----------- , when a = 12 and r = 0.4 d y = r 2 – x 2 , when r = 10 and x = 6 1–r

5B

7 Expand each of the following. a 2(a + 9) b p(2p 4) d 4m5(3m2 2n) e 4xy(4 y)

5B

8 Expand and simplify each of the following. a 2(m + 8) + 6(m + 4) b 3p(p 2) + p(3 p) d 3z(y 2z) + 4y(2y + z) e 4pq(p q) 2p(pq 4)

5C 5C

9 In the formula P = 2l + 2b find l, when P = 78 and b = 24.

c 5p + 8p p f 7x + 4 3x 9 i 4ab + 7a 2b 3ba c 3g2h5 7g2h3 f 32r5s4 ÷ 4r5s i (8m2)2

m ÷ 16m3

c x2(3x3 1) f 6a2b3(2a3 4b2) c 7(2x

4)

3(x + 8)

10 The formula C = 2 r is used to find the circumference of a circle given the radius. Find the radius of a circle with a circumference of 136 m. Give your answer correct to 1 decimal place.

5C 5C

11 In the formula A = 6s2, find s when A = 216.

5D

13 Use the method of substitution to solve the following equations, correct to 1 decimal place. a 5x = 100 b (1.2)x = 2 c (0.75)x = 0.25

5D

14 The amount to which $10 000 will grow when invested at 9% p.a. can be found using the formula A = 10 000 (1.09)n, where n is the number of years of the investment. Use the formula to find the amount of time that it will take for $10 000 to grow into $20 000, correct to the nearest year.

12 The volume of a square-based pyramid can be found using the formula V = 1--- s2h, where s is 3 the side length of the square base and h is the height of the pyramid. Find the side length of a square-based pyramid with a volume of 108.864 cm3 and a height of 6.3 cm.

Chapter 5 Algebraic skills and techniques

189

15 A car depreciates at a rate of 20% p.a. The amount of time that it takes for the car to halve in value can be found by solving the equation (0.8)n = 0.5, where n is the age of the car. Find the length of time it takes for a car to halve in value, correct to the nearest year.

5D

16 Write each of the following in scientific notation. a 600 000 b 0.000 000 000 2 c 78 920 000 000 000 d 0.001 25 e 0.000 004 589 f 124 589 000 000 000

5E

17 Write each of the following in scientific notation, correct to 3 significant figures. a 12 589 b 0.000 125 478 624 c 0.032 143 68 d 586 460 484 e 12 447.151 48 f 0.000 000 051 851 58

5E

18 Write each of the following as a decimal number. a 2.5 102 b 3.87 104 –1 d 2.89 10 e 3.6702 10–7

5E

19 Complete each of the following. a 2.5 105 m = ___ mm c 3.43 104 kL = ___ L e 4.243 107 t = ___ kg

c 9.8504 107 f 1.1 10–3

b 2.8 108 g = ___ kg d 1.45 106 m = ___ km f 1.3 108 mL = ___ L

Practice examination questions 1 multiple choice 3x(2x 4y) 2y(4y A 6x2 – 8y2

6x) = B 6x2 + 8y2

C 6x2 – 24xy – 8y2

D 6x2 – 24xy + 8y2

2 multiple choice The total surface area of a cone is given by the formula A = r(r + s), where r is the radius and s is the slant height of the cone. The formula with s as the subject is: A– r A–r A A A s = ------ – r B s = ---------------C s = -----------D s = ------ + r r r r r 3 multiple choice The total surface area of the square-based pyramid with side of the base b and the height of the triangular face h is given by A = b2 + 2bh. If the total surface area of the pyramid is 64 cm and the length of the side of the base is 4 cm, the height of the triangular face is: A 6 cm B 10 cm C 20 cm D 24 cm 4 multiple choice The solution to the equation 10x = 200 is closest to: A 2 B 2.3 C 2.4

D 20

5E

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Maths Quest General Mathematics HSC Course

5 multiple choice A square has a side length of 5.6 105 cm. The area of the square in scientific notation will be: A 3.136 1011 cm2 B 31.36 1010 cm2 C 3.136 1025 cm2 D 31.36 1025 cm2 6 The volume of a cylinder can be found using the formula V = r 2h. The surface area of a cylinder can be found using the formula SA = 2 r 2 + 2 rh. a Find the volume of a cylinder with a radius of 4.2 cm and a height of 5.5 cm. (Give your answer correct to 1 decimal place.) b Find the height of a cylinder with a volume of 705 cm3 and a radius of 5.2 cm. (Give your answer correct to 1 decimal place.) c Find the radius of a cylinder with a volume of 939.4 cm3 and a height of 7.3 cm (correct to 1 decimal place). d Rewrite the formula for surface area to make h the subject.

CHAPTER

test yourself

5

7 The time taken for an investment to double in value when invested at 7.5% p.a. can be found by solving the equation (1.075)n = 2. a Use the method of substitution to find the solution to this equation, correct to the nearest whole number. b Write an equation that can be used to find the amount of time that it will take for the value of an item to halve in value if it depreciates at 15% p.a. c Solve this equation, correct to 1 decimal place.

Multi-stage events

6 syllabus reference Probability 3 • Multi-stage events

In this chapter 6A Tree diagrams 6B Counting techniques 6C Probability and counting techniques 6D Probability trees

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

6.1

Listing the sample space

6.2

Informal description of chance

6.3

1 List the sample space for each of the following events. a A card is drawn from a standard deck and its suit is noted. b A ball is selected from a bag containing three red, two blue and five white balls. c A pin is stuck in the page of a book and the nearest letter is noted. 2 Describe each of the following events as being certain, probable, fifty–fifty, unlikely or impossible. a Winning the lottery. b Selecting an odd number from cards labelled with numbers 1 to 55. c Finding a $40 note in your wallet. Equally likely events

3 For each of the events in question 2, state whether or not each outcome is equally likely. Fundamental counting principle

6.4

6.5

4 In each of the following find the number of different ways each selection can be made. a One person is to be chosen from each of two classes with 20 people in one class and 25 in the other. b From a menu an entree is to be chosen from a selection of five entrees followed by a main course from a selection of eight, and then a dessert from a selection of six. c Car number plates consisting of two letters, followed by two digits, followed by another two letters. Single event probability

5 Find the probability of each of the following events. a Randomly selecting the winner of a swimming final with eight competitors. b Winning a raffle when 150 tickets are sold and you purchase three tickets. c Selecting a $2 coin from a pocket containing three $2 coins, four $1 coins and seven 20c pieces.

6.6

Determining complementary events

6 Find the complement to each of the following events. a Selecting a vowel from the letters of the alphabet. b Choosing a black marble from a bag with 12 black, 23 white and 15 clear marbles. c Selecting a number less than 10.

6.7

Calculating the probability of a complementary event

7 Find the probability of: a winning a football match given the probability of losing is 2--- . 5 b the train being late given that it is on time four days out of every five. c a golfer missing a putt given the probability of sinking the putt is 0.73.

Chapter 6 Multi-stage events

193

Tree diagrams As discussed in the preliminary course, if an event has more than one stage to it, then it is necessary to draw a tree diagram to list the sample space accurately. In a tree diagram the tree branches out once for each stage of the experiment. At each stage the number of branches is the same as the number of possible outcomes. To list the sample space we then follow the tree to the end of each branch and record the outcome at each stage.

WORKED Example 1

A coin is tossed three times. Draw a tree diagram and use it to list the sample space for this experiment. THINK 1 2 3

WRITE

There are three stages to the experiment. At each stage the outcome can be heads or tails. Draw the tree diagram branching out three times with two branches at each stage.

1st coin

2nd coin Heads

Heads Tails Heads Tails Tails 4

List the sample space by following the path to each end branch.

3rd coin Heads Tails Heads Tails Heads Tails Heads Tails

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, S = {TTT}

To see a step-by-step construction of the tree diagram in worked example 1, click on the PowerPoint icon. In the above example, each stage of the experiment (each toss of the coin) is independent of the other stages. That is to say, the outcome of one toss does not affect the outcome of another toss. In many examples, the outcome of one stage will affect the outcome of another. Consider worked example 2. Here we are forming a two-digit number such that no digit may be repeated. Once a number has been chosen as the first digit, it can not be chosen as the second digit. Therefore, the first stage of the experiment does affect the second stage.

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WORKED Example 2 A two-digit number is formed using the digits 4, 5, 7 and 9 without repetition. Draw a tree diagram and use it to list all possible numbers that can be formed. THINK 1 2

3 4

WRITE

There are two stages to the experiment. For the first stage there will be four branches and since one number is chosen there will be three branches for the second stage. Draw the tree diagram. List the sample space by following the branches to each end point on the tree diagram.

1st digit 4 5 7 9

2nd digit 5 7 9 4 7 9 4 5 9 4 5 7

Sample space 45 47 49 54 57 59 74 75 79 94 95 97

Click on the PowerPoint icon to see this tree diagram constructed step by step. Once we have completed the tree diagram, the probability of an event can be calculated using the formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes

WORKED Example 3 A coin is tossed and a die is rolled. Calculate the probability of tossing a tail and rolling a number greater than 4. THINK 1 2

3

4

5

6

WRITE

There are two stages to the event. At the first stage there are two outcomes and at the second stage there are six outcomes. Draw the tree diagram.

List the sample space by following the branches to each end point on the tree diagram. Calculate the probability using the probability formula. There are two favourable outcomes — T5 and T6. Simplify.

Coin toss Die roll 1 2 3 Heads 4 5 6 1 2 3 Tails 4 5 6

Sample space Heads 1 Heads 2 Heads 3 Heads 4 Heads 5 Heads 6 Tails 1 Tails 2 Tails 3 Tails 4 Tails 5 Tails 6

P(tail and no. > 4) =

2 -----12

P(tail and no. > 4) =

1 --6

Again the PowerPoint icon can be used to see the tree diagram constructed step by step.

Chapter 6 Multi-stage events

195

remember 1. In any probability experiment that has more than one stage, a tree diagram should be used to calculate the sample space. 2. The tree diagram branches once for each stage and the number of branches at each stage is equal to the number of outcomes. 3. The sample space is found by following the path to the end of each branch. 4. Once the sample space has been found, the probability of each outcome is calculated using the probability formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes

6A WORKED

Example

1 A family consists of four children. Draw a tree diagram to show all possible combinations of boys and girls.

3 There are two bags each containing a red, blue, yellow and green marble. One marble is to be chosen from each bag. Draw a tree diagram that will allow you to calculate the sample space.

2

SkillS

Informal description of chance

6.3

SkillS HEET

Example

6.2

HEET

WORKED

SkillS

Listing the sample space

2 Two dice are cast. Draw a tree diagram that will allow you to list the sample space of all possible outcomes.

4 A school is to send one male and one female representative to a conference. The boys nominate George, Frank, Stanisa and Ian; the girls have nominated Thuy, Petria, Joan, Wendy and Amelia. Draw a tree diagram and list the sample space for all possible choices of representatives.

6.1

HEET

1

Tree diagrams

5 A two-digit number is to be formed using the digits 1, 2, 4, 5 and 7 such that no digit Equally likely may be repeated. Draw a tree diagram to list all possible numbers that can be formed. events E

7 The digits 3, 5, 7 and 8 are used to form a three-digit number. If no digit can be used more than once list the sample space. 8 multiple choice From a group of five nominations a school captain and vice-captain are to be elected. The number of ways that the captain and vice-captain can be chosen is: A5 B 10 C 20 D 25 WORKED

Example

3

9 The four aces from a deck of cards are placed face down on a table. One card is chosen followed by a second card without the first card being replaced. Calculate the probability that the ace of hearts is one of the two cards chosen.

sheet

6 A committee needs to elect a president, secretary and treasurer. The four nominations L Spre XCE ad for these positions are Belinda, Dean, Kate and Adrian. Given that no person is allowed to hold more than one position, use a tree diagram to list all ways in which Tree diagrams these three positions can be filled.

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10 A two-digit number is formed using the digits 2, 3, 4 and 7 without repetition. a Use a tree diagram to list the sample space. b Calculate the probability that the number formed is greater than 35. 11 A tennis team consists of three men, Andre, Yevgeny and Jonas and two women, Martina and Lindsay. From the team the captain and the vice-captain are to be chosen. Calculate the probability that the captain and vice-captain are: a Andre and Lindsay b both men c the same sex d different sex. 12 Find the probability that all three children in a family will be the same sex. 13 multiple choice A three-digit number is formed using the digits 5, 6, 8 and 0. No digit can be repeated and the 0 can’t be first. The probability of the number formed being greater than 800 is: A

1 --4

B

1 --3

C

3 -----16

D

1 --2

14 An airline offers holidays to three destinations: Brisbane, Gold Coast or Cairns. The holiday can be taken during two seasons: Peak season or Off-peak season. The customer has the choice of three classes: Economy, Business or First class. There is no First class to Cairns, however. a Use a tree diagram to list all combinations of holiday that could be taken by choosing a destination, season and class. b Terry takes a mystery flight, which means he is allocated a ticket at random from the above combinations. Calculate the probability that Terry’s ticket: i goes to Brisbane ii is First class iii is in Peak season, flying First class.

Counting techniques Ordered arrangements 1 Select three people and stand them in a line. 2 Now get the three people to stand in a different order. 3 In how many different orders can the three people be placed? 4 Repeat the above process with four people in the line. 5 Is there a pattern? Can you calculate the number of different ways in which five people can be arranged?

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There are 10 people standing in a line. In how many ways can they be arranged? To calculate this we need to consider the number of ways that each place in the line can be filled. To do this we need to calculate the number of people remaining after we fill each place in the line. • There are 10 people who could fill the first position. • Once the first position has been filled, there are nine people remaining to fill the second position. • Once the second position has been filled, there are eight people remaining to fill the third position. • This pattern continues until there is only one person left who can fill the last position. Calculating this: 10 9 8 7 6 5 4 3 2 1 = 3 628 800. A shorter way of writing 10 9 8 7 6 5 4 3 2 1 is to write 10!, that is, 10 factorial. Your calculator will have a factorial function, usually labelled x!. Make sure that you know where this function is on your calculator.

WORKED Example 4

Calculate the value of 8!. THINK

WRITE

Enter 8 and press 2ndF calculator.

x! on the

8! = 40 320

Graphics Calculator tip! Factorial function To use the factorial (!) function on your Casio graphics calculator follow the steps below, which will calculate 8! as in worked example 4 above. 1. From the MENU select RUN.

2. Press OPTN (F6) for more options. You should be able to see the screen at right.

3. Press F3 (PROB) and you will be able to see the function x!. Press 8 F1 (x!) EXE .

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WORKED Example 5 Six people are standing in a line. In how many ways can the six people be arranged? THINK

WRITE

1

The answer is 6!.

6! = 720

2

Give a written answer.

The people can be arranged in 720 ways.

Tree diagrams and ordered arrangements Four people, Anji, Belinda, Kristen and Summer, are to be placed in order. 1 Calculate the number of different ways these four girls can be placed in a line. 2 Draw a tree diagram and use it to list the ways that the four girls can be placed in order. 3 Check that the number of elements in the sample space found from your tree diagram corresponds to the answer obtained in part 1. In worked examples 4 and 5, we have been ordering an entire group. In some cases we may wish to order only part of the group. Consider the case of an Olympic swimming final. There are eight swimmers and we wish to know the number of ways that the gold, silver and bronze medals can be awarded. • There are eight possible winners of the gold medal. • With the first place filled, there are seven possible winners of the silver medal. • With both first and second places filled, there are six possible winners of the bronze medal. Calculating this: number of arrangements = 8

7

6

Calculating this: number of arrangements = 336 This type of arrangement is known as an ordered selection. It occurs when the order in which the choices are made is important. In the worked example below, a captain and a vice-captain are to be chosen. If Benito is captain and Imran is vice-captain, this is a different selection to Imran as captain and Benito as vice-captain. To calculate the number of ordered selections that can be made, we multiply, starting from the number of possible first selections, then reducing by one with each multiplication until each position is filled.

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WORKED Example 6 In a cricket team of eleven players, a captain and vice-captain are to be chosen. In how many ways can this be done? THINK 1 2

WRITE

There are 11 possible choices of captain. Once the captain is chosen, there are 10 choices remaining for vice-captain.

No. of arrangements = 11 10 No. of arrangements = 110

Committee selections On a committee of five people, a president and a vice-president are to be chosen. The five committee members are Andreas, Brett, Cathy, Dharma and Emiko. 1 Use the method shown in worked example 6 to calculate the number of ways in which the president and the vice-president can be chosen. 2 Now use a tree diagram to list the sample space of all possible selections of president and vice-president. 3 Check that the number of elements in the sample space corresponds to the answer obtained in part 1 of this investigation. Consider a case where two representatives to a committee are chosen from a class of 20 students. This is an example of an unordered selection. If Sue is chosen, followed by Graham, this is the same choice as if Graham is chosen and then Sue. To calculate the number of unordered selections that can be made, we calculate the number of ordered selections that can be made and then divide by the number of arrangements of these selections. This is calculated using factorial notation as in worked example 5. In the case of choosing the committee: Number of ordered selections is 20 19 = 380. Two people can be arranged in two (2!) ways. Number of unordered selections = 380 ÷ 2 Number of unordered selections = 190

WORKED Example 7 From a group of eight athletes, three are to be chosen to represent the club at a carnival. In how many ways can the three representatives be chosen? THINK 1 2 3

Calculate the number of ordered selections that can be made. Calculate the number of arrangements of the representatives. Divide the ordered selections by the arrangements of the representatives.

WRITE Ordered selections = 8 7 6 Ordered selections = 336 Arrangements = 3 2 1 Arrangements = 6 Unordered selections = 336 ÷ 6 Unordered selections = 56

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Unordered selection A rowing team has six members: Mark, Norman, Olaf, Pieter, Quentin and Raymond. Two are to be chosen to be the crew in a pairs race. 1 Use the method described in worked example 7 to calculate the number of pairs that could be chosen. 2 Use a tree diagram to list the ordered selections and then write the sample space of unordered selections by ignoring any repeated pair. 3 Check that the number of elements of the sample space corresponds to the answer obtained in part 1 of this investigation.

remember 1. A group of n different items can be arranged in n! ways. 2. n! = n (n 1) (n 2) … 1 and can be found as a function on your calculator. 3. When an ordered selection is made, the number of selections can be calculated by multiplying the number of first choices that can be made by the number of second choices that can be made and so on. 4. To calculate the number of unordered selections that can be made, we divide the number of ordered selections by the number of arrangements of those selected.

SkillS

6B HEET

6.4

WORKED

Example

4

Fundamental counting principle

WORKED

Example

5

Counting techniques

1 Use your calculator to calculate the value of the following. a 3! b 5! c 9! 2 Four people are involved in a race. In how many different orders can they complete the race? 3 The letters A, B, C, D and E are written on cards. In how many different orders can the cards be placed? 4 A three-digit number is formed using the digits 3, 6 and 8. If no number can be repeated, how many numbers is it possible to form?

WORKED

Example

6

5 In a race of 10 people, in how many different ways can the first three places be filled? 6 In a school, a captain and vice-captain are to be elected. The four nominations are Geri, Reika, Melanie and Victoria. In how many different ways can the captain and vice-captain be chosen? 7 In the Melbourne Cup there are 24 horses. In how many different ways can the three placings be filled?

Chapter 6 Multi-stage events

WORKED

Example

7

201

8 Seven people try out for three places on a debating team. In how many ways can the team of three be chosen from the group of seven? 9 How many different groups of four can be selected from ten people? 10 In his pocket Trevor has six coins: a $2 coin, $1 coin, 50c coin, 20c coin, 10c coin and 5c coin. If Trevor randomly chooses two coins, how many different sums of money are possible? 11 On a restaurant menu there is a choice of three entrees, six main courses and four desserts. In how many ways can a person choose an entree, main course and dessert from the menu? 12 multiple choice Which of the following is an example of an unordered selection? A Five students are placed in order of their exam results. B From a group of five students, a contestant and a reserve are chosen for a Mathematics competition. C From a group of five students, two are chosen to represent the class on the SRC. D From a group of five students, two are awarded 1st and 2nd prizes in Mathematics. 13 multiple choice The numbers 1, 2, 3 and 4 are used to form a three-digit number such that no digit can be used more than once. The number of three-digit numbers that can be formed is: A4 B 6 C 12 D 24 14 multiple choice Gavin, Dion, Michael, Owen and Shane try out for two places on a tennis doubles team. The number of teams that can be chosen is: A5 B 10 C 20 D 25

16 At the Olympic qualifying trials, nine cyclists compete for a place on the team. a In how many different orders can the competition finish? b How many different ways can 1st, 2nd and 3rd place be filled? c Two cyclists are chosen to represent Australia on the team. How many different teams of two can be chosen?

Probability and counting techniques Once the counting techniques done in the previous section have been completed, we can calculate the probability of certain events occurring. To do this we go back to using the probability formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes

Work

15 A small play has three characters. Six people, Wendy, Rebecca, Thai, Yasmin, Andrea and Ophelia, audition for the three parts. a How many different groups of three can be chosen for the play? b In how many different ways can the three parts be allocated to the three girls? T SHEE

6.1

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WORKED Example 8 The letters A, H, M, S and T are written on cards. The cards are shuffled and then laid out face up. Calculate the probability that the cards form the word MATHS. THINK 1

2

WRITE

The five cards can be arranged in 5! ways.

No. of arrangements = 5! No. of arrangements = 5 4 No. of arrangements = 120 1 P(MATHS) = --------

MATHS is one way of arranging the letters and so we use the probability formula.

3

2

1

120

We also need to be able to calculate the probability of a particular ordered or unordered arrangement occurring.

WORKED Example 9 From Francis, Gary, Harley, Ike and Jacinta, a school captain and vice-captain need to be elected. Calculate the probability that Ike and Jacinta occupy the two positions. THINK 1 2 3 4

WRITE

Calculate the number of ordered selections that are possible. Ike and Jacinta in the two positions can be arranged in two ways. Divide the ordered selections by the number of arrangements. Substitute into the probability formula.

No. of ordered selections = 5 4 No. of ordered selections = 20 No. of arrangements = 2 1 No. of unordered selections = 20 ÷ 2 No. of unordered selections = 10 1 P(Ike and Jacinta) = ----10

WORKED Example 10 A bag contains a red, green, yellow, blue, orange and purple marble. Three marbles are selected from the bag. Calculate the probability that the red, yellow and orange marbles are chosen. THINK 1 2 3 4 5

Calculate the number of ordered selections. Calculate the number of arrangements. Calculate the number of unordered selections. The red, yellow and orange marble is one possible selection. Substitute into the probability formula.

WRITE No. of ordered selections = 6 5 4 No. of ordered selections = 120 No. of arrangements = 3 2 1 No. of arrangements = 6 No. of unordered selections = 120 ÷ 6 No. of unordered selections = 20

P(red, yellow and orange) =

1 -----20

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Popular gaming There are many different forms of lottery that depend upon ordered or unordered arrangements. 1 Lotto — This requires the player to select six numbers out of 45. In how many ways can the six numbers be chosen? Remember order does not matter. 2 Similar games to Lotto are: a Oz Lotto — seven numbers are chosen from 45. b The Pools — six numbers are chosen from 38. In how many ways can the six numbers for each of these games be chosen? 3 Powerball – This requires the player to choose five numbers from 45 in an unordered selection. A sixth ball (the powerball) is chosen from a second barrel containing 45 balls. In how many ways can this be selected? 4 Lotto Strike – The player must select the first four balls drawn from 45 in the correct order. In how many ways can this ordered selection be made?

remember When we have calculated the number of arrangements and the number of ordered or unordered selections that are possible, we can then calculate the probability of a certain selection using the probability formula.

6C WORKED

Example

1 Four people, Craig, Barry, Anne and Dimitri, are arranged in a line. Calculate the probability that the four people are arranged in alphabetical order. 2 The numbers 4, 5, 6, 7 and 8 are arranged to form a five-digit number such that no digit can be repeated. Calculate: a how many five-digit numbers can be formed b the probability that the number formed is 54 867 c the probability that the number formed is 86 574.

Single event probability

3 A three-digit number is formed using the digits 6, 8 and 9 and no digit may be repeated. Calculate the probability that the number formed is: a 896 b even c greater than 800. WORKED

Example

9

6.5

4 There are five candidates in an election for SRC president. The second placed candidate will be made vice-president of the SRC. If Lauren and Meta are two of the candidates, calculate the probability that they will occupy the two positions. 5 Seven surfers enter a competition. If two of the surfers are Kurt and Paul, calculate the probability that: a Kurt comes first and Paul comes second b Paul comes first and Kurt comes second c Kurt and Paul fill the first two places.

SkillS HEET

8

Probability and counting techniques

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6 From the digits 1 to 9 a two-digit number is formed such that no digit can be repeated. Calculate the probability that the number formed is: a 67 b greater than 80 c less than 30. WORKED

Example

10

7 From a deck of cards, the four aces are laid face down on a table. Two of the aces are then turned face up. Calculate the probability that the two aces turned face up are the ace of clubs and the ace of spades. 8 An ice-cream parlour offers a choice of 25 flavours. A triple scoop ice-cream places three different flavours on top of each other. If the flavours are chosen randomly, find the probability that the ice-cream is: a vanilla, chocolate and strawberry in that order b vanilla, chocolate and strawberry in any order. 9 Six boys try out for three places on a debating team. The boys are Gavin, David, Andrew, Rhyse, Julius and Elliot. a How many teams of three is it possible to choose? b Calculate the probability that Gavin, Andrew and Elliot are on the team. 10 The letters M, A, I, D and G are written on cards and two of these are to be chosen. Calculate the probability that the two cards chosen are: a both vowels b both consonants c one vowel and one consonant.

1 1 Two coins are tossed in the air. Use a tree diagram to list the sample space of all possible outcomes. 2 Two dice are rolled. How many possible outcomes are in the sample space? 3 In how many different ways can five cars be parked in a row? 4 A race has 10 runners. In how many different ways can the 10 runners finish? 5 A race has 10 runners. In how many different ways can the first three places be filled? 6 From a committee of nine people, a president and vice-president need to be chosen. In how many different ways can the two positions be filled? 7 Eight people audition for four parts in a play. How many different groups of four could be chosen? 8 Once the four people have been chosen in question 7, in how many different ways can the four parts be allocated? 9 How many different ways can the parts be allocated among the original group of eight in question 7? 10 Explain the difference between an ordered and unordered arrangement.

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Probability trees In the tree diagrams studied so far, the probability of each outcome has been equally likely. When each result is not equally likely we can still draw the diagram in the same way, writing the probability of each single outcome on the branches of the tree. Consider the case where a bag contains three green marbles and two white marbles. A marble is drawn, its colour noted and it is then replaced in the bag. A second marble is then drawn. We could draw a tree diagram as shown on the right.

1st marble Green

Green

Green

White

White

2nd marble Green Green Green White White Green Green Green White White Green Green Green White White Green Green Green White White Green Green Green White White

Using a probability tree simplifies the diagram. In a single drawing of the marble P(green) =

3 --5

and P(white) =

2 --- . 5

These probabilities are drawn on the branches of the

tree as shown below. 1st marble

2nd marble 3– 5

3 – 5

Green

2 – 5 3– 5

2 – 5

White

Green

White Green

2 – 5

White

There are four elements to the sample space: (green, green), (green, white), (white, green) and (white, white). Each element of the sample space is not equally likely. To calculate the probability of each, we use the multiplication rule of probability. The multiplication rule of probability states that to calculate the probability, you multiply along the branches of the tree that lead to each event. Therefore: P(green, green) = = P(white, green) = =

3 --5

3 --5

9 -----25 2 --5 6 -----25

P(green, white) = =

3 --5

P(white, white) = =

3 --5

2 --5

6 -----25 2 --5

2 --5

4 -----25

This is the method that must be used to calculate the probability in any situation where each outcome is not equally likely.

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WORKED Example 11

In a bag there are seven red marbles and three green marbles. A marble is drawn, its colour noted and it is then replaced in the bag. A second marble is then drawn. Find the probability that both marbles are red. THINK 1

WRITE

Draw the probability tree.

1st marble

2nd marble 7 — 10

7 — 10

Red

3 — 10 7 — 10

3 — 10

Green

Red

Green Red

3 — 10

Green 2

Calculate the probability by multiplying along the branches.

P(red, red) = P(red, red) =

7 -----10 49 --------100

7 -----10

The PowerPoint icon will show you step by step how to construct this probability tree. When asked to find the probability of an event that can occur in several ways, we need to use the addition rule of probability. The addition rule for probability states that for an event that can occur in several ways, the probability is the sum of the probabilities for each way that the event can occur.

WORKED Example 12

In a barrel there are four blue counters and six red counters. A counter is drawn, its colour noted and a second counter is drawn. The first counter is not replaced in the barrel before the second counter is drawn. Find the probability that: a a blue counter is drawn, followed by a red counter b two counters of a different colour are drawn. THINK 1

Draw the probability tree. • If the first counter is blue, three blue and six red counters remain in the bag. • If the first counter is red, four blue and five red counters remain in the bag.

WRITE 1st counter 2nd counter 3– 9 4 — 10

Blue

Blue

6 – 9

Red 4– 9 6 — 10

Red

Blue

5– 9

Red

Chapter 6 Multi-stage events

THINK 2

3

207

WRITE

a Multiply along the white, red branches to calculate the probability. b This outcome can occur in two ways. Add the probabilities (blue, red) and (red, blue).

a

P(blue, red) =

4 -----10

=

4 -----15

6 --9

b P(different colour) = P(blue, red) + P(red, blue) 4 = ( ----10

=

4 -----15

=

8 -----15

6 --- ) 9

6 + ( ----10

4 --- ) 9

4 + ----15

The PowerPoint icon will allow you to see how this probability tree was constructed. We must read each example carefully to see if the probabilities change throughout the experiment. In many cases we do not need to examine each possible outcome. In some examples we consider only one outcome. The branches of the tree then show if this outcome occurs or not.

WORKED Example 13 Along a road there are three sets of traffic lights. The probability of catching a green light is 0.35. Calculate the probability of catching all three green lights. THINK 1

WRITE

Draw a probability tree. We do not need to consider if the light is red or amber, only whether it is green or not green.

1st lights

0.35

0.65

2

Calculate the probability by multiplying along the green branches.

2nd lights 0.35

Green

0.65

Not green

Green

Not green

0.35

Green

0.65

Not green

0.35 0.65 0.35 0.65 0.35 0.65 0.35 0.65

P(three green lights) = 0.35 0.35 P(three green lights) = 0.042 875

3rd lights Green Not green Green Not green Green Not green Green Not green

0.35

Click on the PowerPoint icon to see worked example 13 solved step by step. The complementary event method is particularly important with this type of question. Complementary events are two events that account for all possible outcomes of an experiment. For example, when rolling a die the complement of rolling a number less than three is to roll a number greater than two. We discovered during the preliminary course that the sum of the probability of an event and its complement is one. It is often easier to calculate the probability of the complement rather than that of the event itself. We can then subtract the probability of the complementary event from one.

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WORKED Example 14

Three dice are rolled. What is the probability of rolling at least one six?

THINK 1

WRITE

Draw the probability tree. (We need to draw the tree with only two outcomes as we are concerned only with whether we get a 6 or not.)

1st die

2nd die

3rd die 1– 6

1– 6 1– 6

6

5– 6

6 5– 6

Not 6

1– 6

5– 6 1– 6

5– 6

1– 6

6

5– 6

Not 6

5– 6

Not 6

1– 6

5– 6

2 3

The complement to getting at least one six in three rolls is getting no sixes in three rolls. Subtract the complement from one to find the probability.

6 Not 6 6 Not 6 6 Not 6 6 Not 6

P(at least one six) = 1

P(no sixes)

P(at least one six) = 1

( 5--6-

P(at least one six) = 1

125 --------216

P(at least one six) =

5 --6

5 --- ) 6

91 --------216

As with other probability tree diagrams, you can see this example completed step by step by clicking on the PowerPoint icon.

remember 1. If each outcome is not equally likely, draw a probability tree with the probability of each single event on the branches. 2. To calculate a probability, multiply along the branches that give the required outcome. 3. If an outcome can be obtained in two or more ways, add the probability of each. 4. Read each question carefully to see if the probabilities change during the experiment. 5. Consider carefully what outcomes you need to include in your tree. You may need only to consider if one event occurs or not. 6. For questions that involve finding ‘at least one’, use the complementary event method. 7. The sum of the probability of an event and its complement is one.

Chapter 6 Multi-stage events

6D WORKED

Example

Probability trees

1 In a purse there are five 20-cent coins and three 50-cent coins. A coin is selected from the purse and replaced, and then a second coin is selected. The probability tree on the right is drawn for this experiment. Find the probability that the two coins drawn are both twenty cent pieces.

1st coin

2nd coin 5– 8

5– 8

2 In a barrel there are four white marbles and five black 50c marbles. Two marbles are drawn, the first being replaced in the barrel before the second one is drawn. a Draw the probability tree for this situation. b Find the probability for each member of the sample space. WORKED

Example

12

WORKED

Example

13

20c

20c

3– 8

6.6

3– 8

50c

5– 8

20c

3– 8

50c

SkillS HEET

11

209

Determining complementary events

3 A hand of five cards contains three kings and two queens. A card is chosen and then returned before a second card is chosen. Find the probability that: a a queen is chosen followed by a king b a king and a queen are chosen. 4 Jia is a shooter with an 80% chance of hitting a target. If he has three shots at a target, find the probability that: a he hits with all three shots b he hits with exactly two shots.

of a complementary event

6 multiple choice

A bag contains four black and six white marbles. Two marbles are drawn from the bag one after the other. If the first marble drawn is black, the probability that the second marble drawn is white is: A

4 --9

B

2 --5

C

2 --3

D

3 --5

7 multiple choice A coin is biased such that the probability of it landing heads is 0.6. The coin is tossed three times. Which of the following outcomes has the greatest probability of occurring? A Tossing three heads B Tossing two heads and one tail C Tossing one head and two tails D Tossing three tails

SkillS HEET

5 A raffle has 100 tickets with two prizes. Kevin buys five tickets. Find the probability 6.7 that: a Kevin wins 1st prize b Kevin wins both prizes Calculating the probability c Kevin does not win a prize d Kevin wins exactly one prize.

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8 A box contains three red and seven blue discs. Two discs are chosen from the box. The probability tree for this experiment is shown on the right. Find the probability of selecting: a two red discs b two blue discs c two discs the same colour d two discs of a different colour.

1st disc

3 — 10

7 — 10

2nd disc 2 – 9

Red

7 – 9

Blue

3 – 9

Red

6 – 9

Blue

Red

Blue

9 The names of eight boys and five girls are placed into a hat. Two people selected from the hat are to represent the school at a function. a Use a probability tree to find the sample space for this experiment. b Find the probability of: i two boys being chosen ii two girls being chosen iii one boy and one girl being chosen.

10 There are 25 students in class 12R and 24 students in class 12S. Two students are to be chosen at random to attend a study skills course. Find the probability that the two students chosen are: a from the same class b from different classes. 11 In a basket there are 15 balls, of which five are blue. Two are selected at random from the basket. Find the probability that: a two blue balls are selected b no blue balls are selected c exactly one blue ball is selected. 12 The probability that I will need to stop at a set of traffic lights is 0.55. If I twice travel through this set of lights, what is the probability of: a having to stop both times b not having to stop either time. 13 Greg has an 80% chance of passing each Maths test. During the term he will need to sit four tests. 14 a Find the probability that Greg will pass all four tests. b Find the probability that Greg will fail at least one test.

WORKED

Example

14 A navy ship carries surface-to-air missiles with a probability of hitting a target of 0.9. Two missiles are fired at an enemy warplane. Find the probability that the warplane escapes without being hit. (Hint: For the plane to escape, both missiles that are fired must miss the target.) 15 In a certain town it is known that four-fifths of all school students have been immunised against measles. For a medical test, four students need to be chosen of which at least one must have been immunised and at least one must not have been immunised. Find the probability that if four students are chosen at random: a at least one will have been immunised b at least one will not have been immunised.

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16 multiple choice Veronica rolls three dice. To win the game she needs to throw at least one six. Which of the following will give the probability of throwing at least one six? A 1 − P(three sixes) B 1 − P(two sixes) C 1 − P(one six) D 1 − P(no sixes) 17 There are 2 classes in Year 12: Class 12A has 15 boys and 10 girls. Class 12B has 12 boys and 18 girls. The principal chooses a student to make a speech by first choosing a class at random followed by a student at random from the chosen class. Find the probability that the student chosen is: a from class 12A b a boy from class 12B c a girl.

19 A missile that is fired from the ground has a 0.8 chance of hitting its target. A missile fired from a plane has a 0.4 chance of hitting a target. A missile is fired from both ground and air at separate targets. Find the probability that: a both hit their target b one hits its target c at least one hits its target.

Work

18 In a radio contest, to win $10 000 in prize money the contestant is faced with five money bags. Each money bag has 10 coins in it. To win, the contestant chooses a bag and then chooses a coin from that bag. If the coin has the station logo on it, the contestant wins. Bag 1 has one winning coin. Bag 2 has three winning coins. Bag 3 has seven winning coins. Bags 4 and 5 have two winning coins. Find the probability of the contestant winning the $10 000.

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summary Tree diagrams • A tree diagram is used in any probability experiment where there is more than one stage to the experiment. • The sample space can be determined from a tree diagram by following the paths to the end of each branch. • The probability of an event can then be calculated by the probability formula: number of favourable outcomes P(event) = ---------------------------------------------------------------------------total number of outcomes

Counting techniques • The number of ways that n objects can be arranged in order is: n! or n (n 1) (n 2) . . . 2 1. • In an ordered selection, a number of objects are chosen and are arranged in order. The number of ordered selections can be calculated by multiplying the number of first choices that can be made by the number of second choices possible and so on until all choices have been included. • In an unordered selection, the order in which the objects have been chosen is not important. The number of unordered selections that are possible is calculated by dividing the number of ordered selections by the number of ways the ordered selection can be arranged. • Once the number of selections has been determined, the probability of particular selections can be determined.

Probability trees • When each outcome is not equally likely, you draw a probability tree. • On each branch of the tree is written the probability of that outcome. • To calculate any probability you multiply along the branches.

Chapter 6 Multi-stage events

213

CHAPTER review 1 Two coins are tossed in the air. a Draw a tree diagram. b Use the tree to list the sample space for this experiment.

6A

2 The digits 5, 7, 8 and 9 are used to form a two-digit number. Use a tree diagram to list the sample space if: a no digit can be used more than once b digits can be repeated.

6A

3 There are three births in the maternity ward of a hospital. Calculate the probability that the babies are: a all boys b two boys and a girl c more girls than boys.

6A

4 A two-digit number is formed using the digits 4, 6, 7, 8 and 9. No digit is allowed to be repeated. a Use a tree diagram to list the sample space. b Find the probability that the number formed is: i 86 ii odd iii greater than 65.

6A

5 In a barrel there are three black marbles and three white marbles. A marble is drawn and its colour noted, and it is then replaced in the barrel. A second marble is then drawn. Find the probability of selecting: a two marbles of the same colour b at least one black marble.

6A

6 A rowing crew has eight rowers. In how many different ways can the crew be seated in the boat?

6B 6B

7 From the rowing crew of eight, a captain and vice-captain are to be selected. Calculate the number of different ways the captain and vice-captain can be selected. 8 From the rowing crew of eight, four are to be chosen to crew a four-person boat. How many crews of four can be chosen from the group of eight?

6B

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9 From the digits 1, 2, 3, 4 and 5: a how many five-digit numbers can be formed if repetition is not allowed? b how many three-digit numbers can be formed if repetition is not allowed?

6C

10 The letters D, S, T, U and Y are shuffled and placed in a line on a table. Calculate the probability that the word STUDY is formed.

6C

11 Two students from Richard, Sandra, Talia and Ingo have to make a speech. They draw straws to see who will go first and second. a How many different ways can the first and second speaker be arranged? b What is the probability that Ingo speaks first and Talia speaks second?

6C

12 Six teams A, B, C, D, E and F contest a basketball competition. The top four sides play in the semi-finals, and later two will contest the grand final. a In how many different ways can the top four sides be arranged? b What is the probability that the top four teams finish D, C, F and A? c How many pairs of teams is it possible to meet in the grand final? d What is the probability of A playing B in the grand final? e What is the probability that C plays in the grand final?

6C

13 Zita is doing an exam when she realises that she has almost run out of time. She has not answered the last 10 questions. a If each question requires True or False as an answer and Zita guesses each answer, what is the probability that she guesses all 10 correctly? b If each question is multiple choice and requires the choice of (A), (B), (C) or (D), what is the probability that Zita will guess all 10 correctly?

6D

14 In a bag there are three red marbles and two green marbles. Two marbles are drawn in succession without replacement. Find the probability that the two marbles drawn are: a both red b both green.

6D

15 In a box there are six batteries. Two of the batteries are flat. If two are chosen from the box, find the probability that both batteries are charged.

6D

16 The probability that a set of lights show green is 2--- . If I pass through this set of lights three 5 times, find the probability that: a I catch three green lights b I catch at least one green light.

6D

17 In a tennis match it is noticed that Roger Federer gets 70% of serves in play. If he has two serves, find the probability that he gets at least one into play.

6D

18 One in every eight light bulbs are faulty. If I buy three light bulbs, find the probability that none are faulty.

Chapter 6 Multi-stage events

Practice examination questions 1 multiple choice Which of the following is an example of an ordered selection? A A team of four people is chosen from a group of 12. B Two representatives from a class of 30 students are elected to the SRC. C A class of 30 students elect a class captain and vice-captain. D From a barrel of 44 balls, six are chosen. 2 multiple choice Six people are arranged in a line. The number of ways in which this can be done is: A 6 B 12 C 120 D 720 3 multiple choice In a race there are six runners. In how many ways can the first three places be filled? A 6 B 12 C 120 D 620

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4 multiple choice A group of six people consists of Darren, Shintaro, Jim, Damien, John and Allan. From these six people a group of three is chosen. The probability of choosing Darren, Jim and John is: A

3 --6

B

1 -----20

C

1 -----12

D

1 --------120

5 Three coins are tossed in the air. a Draw a tree diagram to list the sample space. b Use your tree diagram to calculate the probability of tossing two heads and one tail. c Calculate the probability of tossing at least one head.

CHAPTER

test yourself

6

6 A basketballer has a probability of 0.4 of landing a three point shot. The basketballer has two shots at the basket. a Draw a probability tree showing all possible results of the two shots. b Calculate the probability that the basketballer: i lands both shots ii lands exactly one shot iii lands at least one shot.

Applications of probability

7 syllabus reference Probability 4 • Applications of probability

In this chapter 7A Expected outcomes 7B Financial expectation 7C Two-way tables

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

7.1

Single event probability

7.2

Tree diagrams

7.3

Probability trees

1 Calculate the probability of each of the following. a Rolling a die and getting a number greater than 2. b Winning a raffle after purchasing 10 tickets and knowing there are 500 tickets in the draw. c Selecting an even number from the numbers 1 to 99 inclusive.

2 Two of the digits 3, 5, 6 and 7 are used to form a two-digit number such that no digit can be repeated. Draw a tree diagram to list all possible two-digits numbers that can be formed.

3 In any given hour of television there are 12 minutes of advertisements. If Tony turns the television on at two randomly selected times between 7.00 pm and 8.00 pm. a use a probability tree to show all possible outcomes b calculate the probability that on both occasions Tony turns on the television during an advertisement.

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Expected outcomes Suppose that we toss a coin 100 times. How many times would you expect the coin to land Heads? As each outcome is equally likely, we would expect there to be 50 Heads and 50 Tails. How can this be shown to be true? The number of times that we expect a certain outcome to occur is found by multiplying the probability of each outcome by the number of trials. In the above case, the probability of the coin landing heads is --1- , and this is multiplied by the number of trials 2

(100). The result is an expectation of 50 Heads in 100 tosses of the coin. The expected outcome is the number of times that we expect a particular outcome to occur in a certain number of trials.

WORKED Example 1

A die is rolled 120 times. How many 6s would you expect to occur in 120 rolls of the die? THINK 1 2

WRITE

Calculate the probability of rolling a 6. Multiply the probability of a 6 by the number of trials.

P(six) = 1--6 Expected number of 6s =

1 --6

120

Expected number of 6s = 20

If the expected number of 6s is 20 in 120 rolls of a die, this does not mean that this is what will occur. It may be that on one occasion we may get 25 sixes in 120 rolls, another occasion we may get only 10 sixes. However, we expect that if we repeat the experiment often enough, we would get an average of 20 sixes in 120 rolls.

Rolling a die 1 Each person is to take a die and roll it 120 times and record the number of 6s rolled. 2 What is the most number of 6s rolled by anyone in 120 rolls of the die? 3 What is the least number of 6s rolled by anyone in 120 rolls of the die? 4 What is the average number of 6s rolled by the class in 120 rolls of the die? How does this compare with the expected outcome of 20?

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The expected outcome does not need to be a whole number. In many cases this will not be so. Consider the example below.

WORKED Example 2 Roger draws a card from a standard deck, notes the suit and replaces the card in the deck. If Roger repeats this process 50 times, how many spades can Roger expect to have drawn? THINK 1

2

WRITE

Calculate the probability of drawing a spade. Calculate the expected number of spades by multiplying the probability by the number of trials.

P(spade) =

1 --4

Expected number of spades =

1 --4

50

Expected number of spades = 12.5

Obviously, after drawing 50 cards, Roger could not have drawn 12.5 spades. The number of spades drawn must of course be a whole number. However, if this experiment were repeated a number of times, we would expect to have drawn an average of 12.5 spades in every 50 cards. The expected outcome method can be applied to any probability experiment. This includes multistage events in which it may be necessary to draw a tree diagram or probability tree to calculate the probability of a particular outcome.

WORKED Example 3 A psychologist is conducting a study on the upbringing of boys. For the study, the psychologist selects 100 couples with exactly three children. How many of these couples would the psychologist expect to have three boys? THINK 1

WRITE

Draw a tree diagram showing the sample space for three children.

Boy Boy Girl Boy Girl Girl

2 3

Calculate the probability of three boys. Calculate the expected number by multiplying the probability of three boys by the number of couples in the study.

Boy Girl Boy Girl Boy Girl Boy Girl

P(three boys) = 1--8 Expected number of couples with three boys =

1 --8

100

= 12.5

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remember 1. The number of times an event can be expected to occur in a number of trials is calculated by multiplying the probability of that event by the number of trials. 2. The number of times we expect an event to occur does not mean the event will occur that number of times. Rather, this is the average number of times we would expect this event to occur.

7A WORKED

Example

1 Calculate the number of times that a coin can be expected to land Tails in 40 tosses. 2 A die is rolled 300 times. Calculate the expected number of 6s to be rolled. 3 A card is drawn from a standard deck, its suit is noted and the card is replaced in the deck. Calculate the expected number of hearts in 100 selections.

7.1 Single event probability

7.2

2

Tree diagrams

7.3

SkillS

HEET

Example

5 Lorna spends a night at the greyhounds. There are 10 races, and in each race there are eight greyhounds. Lorna bets on number 5 in every race. Calculate the number of winning greyhounds that Lorna can expect to back.

SkillS

HEET

4 A barrel contains five red marbles, four blue marbles and a green marble. A marble is drawn from the barrel. Its colour is noted, and it is then replaced in the barrel. In 70 selections from the barrel, how many times would we expect to select: a a red marble? b a blue marble? c a green marble? WORKED

SkillS

HEET

1

Expected outcomes

Probability trees

6 A card is drawn from a standard deck; the card is then noted and replaced in the deck. This is repeated 100 times. Calculate the number of times (where necessary, correct to 2 decimal places) that we could expect to select: a a club b a red card c an ace d a court card (ace, king, queen or jack) e the king of diamonds. 7 Kevin buys a ticket in a meat raffle every week. There are 100 tickets and four prizes. a Calculate the probability of Kevin winning a prize in the raffle. b How many prizes can Kevin expect to win in one year? 8 Janice buys a ticket in every lottery. In each lottery there are 180 000 tickets, a first prize and 3384 cash prizes. One lottery is drawn every weekday for 52 weeks a year. Calculate the number of times in 10 years that Janice can expect to win: a first prize (as a decimal, correct to 3 significant figures) b a cash prize (as a decimal, correct to 3 significant figures).

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9 multiple choice A meeting is attended by 350 men and 150 women. At the meeting 100 people will be chosen to make a speech. What is the expected number of women to make speeches? A 15 B 30 C 50 D 150 10 multiple choice A tennis club runs a raffle each week with 100 tickets. Fumiko buys one ticket each week. The expected number of raffle wins over a period of 50 weeks is: A 0.01 B 0.5 C1 D 20 11 Four coins are tossed simultaneously in the air. If this were repeated 80 times, on how many occasions would you expect the coins to land with four Heads? 3 12 The digits 2, 5, 6, 7 and 9 are written on cards and placed face down. Three are then chosen and arranged to form a three-digit number. If this is repeated 150 times, what is the expected number of: a odd numbers? b numbers greater than 600? c multiples of five?

WORKED

Example

EXCE

et

reads L Sp he

Die rolling

13 Two dice are rolled 100 times. Copy and complete the table below to calculate the expected number of occurrences of each total in 100 rolls of the dice. Give each answer correct to 1 decimal place. Outcome

2

3

4

5

6

7

8

9

10

11

12

Probability Expected no. 14 A barrel contains 15 blue marbles and 5 red marbles. Two marbles are selected from the barrel, the first not being replaced in the barrel before the second is chosen. This experiment is repeated 100 times. On how many occasions (correct to 2 decimal places) would you expect the two marbles chosen to be: a both blue? b both the same colour? c different colours? d selected with at least one being blue?

Computer Application 1 Simulations EXCE

et

reads L Sp he

Simulations

A simulation is where a computer gives results to an experiment that are similar to those that would occur if the experiment were actually performed. For example, if a coin is tossed 100 times, a computer can randomly choose Heads or Tails in a fraction of a second. In each case, the probability of each outcome is 1--- and we are saved the 2 process of actually tossing the coin. 1. Access the spreadsheet Simulations from your Maths Quest General Mathematics HSC Course CD-ROM.

Chapter 7 Applications of probability

223

2. The first worksheet has a coin toss simulation. In cell B3 enter the number of times you wish to toss the coin, in cell F4 enter the expected number of heads and in cell F5 enter the expected number of tails. 3. How do the simulation results compare with the expected outcome? Complete 10 simulations and average the results. Is this answer closer to the expected number of outcomes that you have calculated? 4. Repeat this process for each one of the other simulations on rolling a die and rolling two dice.

Financial expectation We can use expected outcomes to make an assessment of financial situations where probability is concerned. In particular, this applies to many forms of gambling. The average financial outcome from such a situation is called the financial expectation. Consider a simple game where two people are betting $1 on the toss of a coin. The probability of winning the toss is

1 --2

and this will give a financial return of $1, while the

probability of losing the game is

1 --2

and this will lead to a financial loss of $1. We need

to consider a financial loss as being negative. To calculate the financial expectation, we multiply each financial outcome by the probability of that outcome and then add the results together. In the above example: Financial expectation =

1 --2

= $0

$1 +

1 --2

$1

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This financial expectation tells us that we can expect to neither gain nor lose money in this game over a long period of time. This does not mean that this will be the outcome, but it is the average expected outcome.

WORKED Example 4 A game is played where a die is rolled. If a six is rolled, the player wins $6; if a five is rolled, the player wins $3; and if any other number is rolled, the player loses $3. What is the financial expectation from this game? THINK

WRITE

Financial expectation is calculated by multiplying the financial result of each outcome by the probability of each outcome and adding the results together.

Financial expectation =

1 --6

Financial expectation =

$3 + $3 +

1 --6

$3 +

1 --6

$3 +

Financial expectation = $3 + 1--6 Financial expectation = $0.50

1 --6

1 --6

$6

In worked example 4, the financial expectation is negative. This means that over an extended period of time we can expect to lose 50c per game. This type of calculation can be applied to other financial situations such as the share market.

WORKED Example 5 Over the past 10 years the price of a particular share has risen by $2 on five occasions, by $1 on two occasions and has fallen by $3 on three occasions. What is the financial expectation for this share price in the next year? THINK 1

2

WRITE 5 ------ , 10

Calculate the (experimental) probability of each outcome.

P($2 profit) =

Calculate the financial expectation using the experimental probabilities.

Financial expectation =

P($3 loss) =

P($1 profit) =

2 ------ , 10

3 -----10 5 -----10

Financial expectation = +

$2 + 3 -----10

2 -----10

$1

$3

Financial expectation = $0.30 In this example, where the financial expectation is positive, we can expect to make a profit. Again this does not mean we will make a profit but the average share price fluctuation is a gain of 30c.

remember 1. Financial expectation is the average return in a financial situation. 2. The financial expectation is calculated by multiplying each possible financial outcome by the probability of that financial outcome and adding the results together. 3. A financial loss is indicated by a negative financial outcome while a financial gain is a positive financial outcome.

Chapter 7 Applications of probability

7B WORKED

Example

4

225

Financial expectation

1 A game is played where a die is rolled. If a 1 or a 6 is rolled, the player wins $2; if any other number is rolled, the player loses $1. What is the financial expectation from this game? 2 There are five cards labelled 1, 2, 3, 4 and 5. A card is selected. If it is even, you win $5, and if it is odd, you lose $4. Calculate the financial expectation. 3 Soon-Jung plays a game in which two coins are tossed. If he throws two Heads, he wins $5; if he throws two Tails, he loses $3. For one Head and one Tail, he loses $2. Calculate the financial expectation from this game. 4 In a card game, the player selects a card from a standard deck. The player then wins $5 for an ace and $2 for a king, queen or jack. If any other card is selected, $1 is lost. Calculate the financial expectation from this game. 5 A raffle has 1000 tickets that sell for $1 each. There is a first prize of $400, a second prize of $200 and a third prize of $100. Calculate the financial expectation from the purchase of one ticket in the raffle.

WORKED

Example

5

6 Over the past 20 years shares in the company FIA have increased by $5 on eight occasions, increased by $2 on six occasions and fallen by $3 on six occasions. Calculate the financial expectation for a person who buys FIA shares for the coming year. 7 Look at the roulette wheel on the right. a How many slots are on the wheel? b How many of these slots are: iii black? iii red? iii green? c Francis bets $10 on black. If a black number is spun, he wins $10; otherwise, he loses $10. Calculate Francis’s financial expectation. 8 multiple choice A game is played where a die is rolled. The cost of the game is $1. The players are returned their $1 plus an extra $5 if they can roll a 6. The financial expectation from this game is: A0 B 0.17 C −0.17 D −1 9 multiple choice Which of the following games has the greatest financial expectation? A A coin is tossed. Players win $1 if they toss a Head; otherwise, $1 is lost. B Two coins are tossed. Players win $2 if they toss two Heads; otherwise, $1 is lost. C A die is rolled. The player wins $6 for rolling a 6; otherwise, $1 is lost. D Two dice are rolled. The player wins $6 for rolling a total of six; otherwise, $1 is lost.

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Work

10 In a dice game, two dice are rolled. • The player wins $1 for rolling a total of 7 or 11. • The player loses $1 for rolling a total of 2, 3 or 12. • If any other total is rolled, the dice are rolled again. What is the financial expectation from this game?

T SHEE

7.1

11 In the Jackpot lottery there are 180 000 tickets sold at $2 each. The prizes are shown below. 1st prize $100 000 2nd prize $10 000 3rd prize $5000 2 prizes of $1000 2 prizes of $500 5 prizes of $200 12 prizes of $100 60 prizes of $50 600 prizes of $20 2700 prizes of $10 Calculate the financial expectation from purchasing a $2 lottery ticket.

1 1 Calculate the expected number of sixes in 120 rolls of a die. Information for questions 2 to 5. A pack of cards is shuffled, a card is chosen and then returned to the deck. The cards are then shuffled again. If this process is repeated 100 times, calculate (correct to 1 decimal place) the expected number of: 2 clubs 3 red cards 4 kings 5 court cards. 6 A game is played where a die is rolled. The player wins $3 for a six, $2 for a five and loses $1 for any other result. Calculate the financial expectation for this game. 7 A game is played where two dice are rolled. The player wins $20 for a total of 12, $10 for a total of 2 and loses $1 for any other total. Calculate the financial expectation for this game. 8 A game is played where the financial expectation is 0.2. Explain what this means. 9 A game is played where the financial expectation is –0.2. Explain what this means. 10 Over the past 10 years the share price in a company has risen by $5 in three of the years and has fallen by $1.50 in the other seven years. Based upon these results, if I purchase shares in this company, what would be my financial expectation for the year ahead?

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Two-way tables A two-way table is a two-dimensional grid that shows the outcome of an experiment in terms of two variables. A two-way table is used to display information and allows for predictions to be made based on this information. Consider an example where 400 newborn babies are tested for a genetic condition. The two-way table below displays these results. Test results Accurate

Not accurate

Total

85

9

94

Without condition

304

2

306

Total

389

11

With condition

The information that is given to us by this two-way table is that: • 94 babies have the condition of which 85 were diagnosed and 9 were not • 306 babies did not have the condition of which 304 were shown not to have the condition by the test and 2 who were told they had the condition but they did not (these are known as false positives). From such a two-way table we can tell the total number of babies with and without the condition and the total number of correct and incorrect diagnoses made.

WORKED Example 6 A new test was designed to assess the reading ability of students entering high school. The results were used to determine if the students’ reading level was adequate to cope with high school. The students’ results were then checked against existing records. • 150 adequate readers sat for the test and 147 of them passed. • 50 inadequate readers sat for the test and 9 of them passed. Present this information in a two-way table. THINK

WRITE

Draw up the table showing the number of students whose reading was adequate and the number of students for whom the results of the new test were confirmed.

Test results

Adequate readers Inadequate readers Total

Passed

Did not pass

Total

147

3

150

9

41

50

156

44

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When information is presented in a two-way table, conclusions can be made about the accuracy of such a test and calculations can be made about the probability that such a test is accurate.

WORKED Example 7

A batch of sniffer dogs is trained by customs to smell drugs in suitcases. Before they are used at airports they must pass a test. The results of that test are shown in the two-way table below. Test results Detected

Not detected

Total

No. of bags with drugs

24

1

25

No. of bags without drugs

11

164

175

Total

35

165

a b c d

How many bags did the sniffer dogs examine? In how many bags did the dogs detect drugs? In what percentage of bags without drugs did the dogs incorrectly detect drugs? Based on the above results, what is the probability that the dogs will not detect a bag carrying drugs?

THINK

WRITE

a Add both total columns; they should give the same result.

a 200 bags were examined.

b The total of the detected column.

b The dogs detected drugs in 35 bags.

c There were 175 bags without drugs but dogs incorrectly detected them in 11 bags. Write this as a percentage.

c Percentage incorrectly detected

d Use the probability formula. Of 25 bags with drugs, 1 went undetected.

d P(bag going undetected) =

=

11 --------175

100%

= 6.3% 1 -----25

As a result of studying a two-way table, we should also be able to make judgements about the information given in the tables. In the above worked example only one bag out of 25 with drugs went undetected. Although the dogs incorrectly detected drugs in 11 bags that did not have them, they still have an overall accuracy of 94%. Many two-way tables will require you to make your own value judgements about the conclusions established by the test. For example, the 94% overall accuracy recorded above may be considered very acceptable.

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Chapter 7 Applications of probability

remember 1. A two-way table is used to display test results and examine the accuracy of these results. 2. The table displays horizontally the numbers with and without a certain condition, and vertically displays information about accuracy. 3. The table can be used to make calculations about the accuracy of the test and about the probability of those test results being accurate in an individual case.

7C

Two-way tables

1 A test is developed to test for the flu virus. To test the accuracy, the following 500 L Spre XCE ad people are tested. 6 • 100 people who are known to have the flu are tested and the test returns 98 positive Two-way frequency results. tables • 400 people who are known not to be infected with the virus are tested with 12 false positives being returned.

WORKED

E

Display this information in the two-way table below. Test results Accurate

Not accurate

Total

With virus Without virus Total 2 One thousand people take a lie detector test. Of 800 people known to be telling the truth, the lie detector indicates that 23 are lying. Of 200 people known to be lying, the lie detector indicates that 156 are lying. Present this information in a two-way table. 3 The two-way table shown below displays the information gained from a medical test screening for a virus. A positive test indicates that the patient has the virus. 7 Test results

WORKED

Example

Accurate

Not accurate

Total

45

3

48

Without virus

922

30

952

Total

967

33

With virus

a How many patients were screened for the virus? b How many positive tests were recorded? (that is, in how many tests was the virus detected?) c What percentage of test results were accurate? d Based on the medical results, if a positive test is recorded, what is the probability that you actually have the virus?

sheet

Example

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Maths Quest General Mathematics HSC Course

4 The two-way table below indicates the results of a radar surveillance system. If the system detects an intruder, an alarm is activated. Test results Alarm activated Intruders No intruders Total

Not activated

Total

40

8

48

4

148

152

44

156

a Over how many nights was the system tested? b On how many occasions was the alarm activated? c If the alarm was activated, what is the probability that there actually was an intruder? d If the alarm was not activated, what is the probability that there actually was an intruder? e What was the percentage of accurate results over the test period? f Comment on the overall performance of the radar detection system. The information below is to be used in questions 5 to 7. A test for a medical disease does not always produce the correct result. A positive test indicates that the patient has the condition. The table indicates the results of a trial on a number of patients who were known to either have the disease or known not to have the disease. Test results Accurate

Not accurate

Total

57

3

60

Without disease

486

54

540

Total

543

57

With disease

5 multiple choice The overall accuracy of the test is: A 90%

B 90.5%

C 92.5%

D 95%

6 multiple choice Based on the table, what is the probability that a patient who has the disease has it detected by the test? A 90%

B 90.5%

C 92.5%

D 95%

Chapter 7 Applications of probability

231

7 multiple choice Which of the following statements is correct? A The test has a greater accuracy with positive tests than with negative tests. B The test has a greater accuracy with negative tests than with positive tests. C The test is equally accurate with positive and negative test results. D There is insufficient information to compare positive and negative test results. 8 Airport scanning equipment is tested by scanning 200 pieces of luggage. • Prohibited items were placed in 50 bags and the scanning equipment detected 48 of them. • The equipment detected prohibited items in five bags that did not have any forbidden items in them. a Use the above information to complete the two-way table below. Test results Accurate

Not accurate

Total

Bags with prohibited items Bags with no prohibited items

b Use the table to answer the following: i What percentage of bags with prohibited items were detected? ii What was the percentage of false positives among the bags that had no prohibited items? iii What is the probability of prohibited items passing through the scanning equipment undetected? iv What is the overall percentage accuracy of the scanning equipment?

Work

Total

T SHEE

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summary Expected outcomes • The expected number of times that an event will occur in a number of trials is calculated by multiplying the number of trials by the probability of that event occurring. • The expected number of outcomes is the average number of times that the event is expected to occur. It does not mean this is the number of times the event will occur.

Financial expectation • Financial expectation is the average financial position at the end of a situation where either a profit or loss will be made. • The financial expectation is calculated by multiplying each possible financial outcome by the probability of that outcome and then adding the results together.

Two-way tables • A two-way table is used to display the results of a test and assesses the accuracy of such a test. • The table can be used to calculate the overall probability of the test achieving its objectives.

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233

CHAPTER review 1 Thirty-six coins are tossed in the air. Calculate the expected number of coins landing Heads. 2 A die is rolled 60 times. Calculate the expected number of: a 6s b even numbers

c numbers less than 3.

7A 7A

3 A card is chosen from a standard deck, noted and replaced in the deck. In 100 trials, calculate (where necessary, correct to 2 decimal places) the expected number of: a red cards b spades c aces d court cards e black jacks.

7A

4 Two dice are rolled. The score in each roll is the total of the two dice. In 90 rolls of the dice, calculate the expected number of: a twos b sevens c tens d doubles e totals greater than 8.

7A

5 In a game, three coins are tossed in the air. In 100 tosses of the coins, on how many occasions would you expect the coins to land: a three Heads? b two Tails and one Head? c more Heads than Tails?

7A

6 Two-digit numbers are formed using the digits 2, 4, 7 and 8, and no digit may be repeated. If 60 such numbers are formed, how many numbers can be expected to be: a 47? b even? c less than 40?

7A

7 Alex bets $10 on the toss of a coin. He calls Heads. If the coin lands Heads, Alex wins $10; if it lands Tails, he loses $10. What is his financial expectation?

7B

8 A bag contains 10 marbles, each with an amount of money written on it. Five marbles have $1 written on them, two have $2 written on them and the others have $5, $10 and $20 written on them. A player pays $5 to draw a marble from the bag and is then returned the amount of money on the marble that is drawn. Calculate the financial expectation from this game.

7B

9 Explain the difference between a positive and negative financial expectation.

7B 7B

10 A roulette wheel is spun (see photograph page 210). Carly bets $1 on number 25. If 25 is the number spun, Carly will win $35 and have her $1 returned; if not, she will lose $1. Calculate the financial expectation from this game. 11 Jason plays a game where he rolls two dice. If he rolls a total greater than 9, he wins $5; otherwise, he loses $1. Calculate the financial expectation from this game.

7B

12 A bag contains 20 marbles of which 10 are black, 9 are white and 1 is red. Kerry draws a marble from the bag at random. If a black marble represents a $5 loss, a white marble a $4 gain and a red marble a $20 gain, calculate the financial expectation from this game.

7B

13 Over the past 15 years the share price of PHB has risen by $4 in 12 of the years, fallen by $5 in two years and fallen by $10 in the others. If I buy shares in PHB, what would my financial expectation be for the coming year?

7B

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14 A medical test screens 200 people for a virus. A positive test result indicates that the patient has the virus. • Of 50 people known to have the virus, the test produced 48 positive results. • Of the remainder who were known not to have the virus, the test produced one positive result. Use the above information to complete the table below. Test results Accurate

Not accurate

Total

With virus Without virus Total

7C

15 The results of a lie detector test are given below. • Of 80 people known to be telling the truth, the lie detector indicates that three are lying. • Of 20 people known to be lying, the lie detector indicates that 17 are lying. Display this information in a two-way table.

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16 Below are the results of a test screening for a disease. A positive test indicates that the patient has the disease.

7C

Test results Accurate

Not accurate

Total

18

2

20

Without disease

108

12

120

Total

126

14

With disease

a How many people were tested for the disease? b How many positive test results were recorded? c What percentage of those people with the disease were correctly diagnosed by the test? d If a person without the disease is chosen at random, what is the probability that they returned a positive test? 17 A reading test for people with dyslexia is given and the results are shown in the two-way table below. Test results Accurate

Not accurate

Total

With dyslexia

39

1

40

Without dyslexia

85

5

90

124

6

Total

a How many people were tested? b What percentage of people tested positive to dyslexia? c Based on the above results, if a person with dyslexia takes the test, what is the probability that they will be accurately diagnosed?

Practice examination questions 1 multiple choice A bag contains 3 red marbles, 13 blue marbles and 4 yellow marbles. A marble is chosen from the bag and then replaced in the bag. In 90 selections, the expected number of blue marbles selected is: A 13 B 20 C 58.5 D 59 2 multiple choice A game is played where the player tosses four coins in the air. If all four coins have the same face up, the player wins $6. Otherwise the player loses $1. The financial expectation from this game is: A −$1.00 B −$0.125 C $0.125 D $6.00

7C

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3 multiple choice The two-way table below shows the results of a trial on new metal detectors for aircraft. The metal detector scans a piece of hand luggage and lights up if metal is found. Test results Accurate

Not accurate

Total

9

1

10

Without metal

87

3

90

Total

96

4

With metal

Based on the above results, the probability of metal going undetected in a piece of hand luggage is: A 10% B 25% C 75% D 90% 4 A game is played where two dice are rolled. a Calculate the probability of rolling a total of 7. b How many times would you expect to roll a total of 7 in 90 rolls of two dice? c Calculate the probability of rolling a total of 11. d Xiao plays a game where he wins $3 for rolling a total of 7 and $7 for rolling a total of 11. Otherwise he loses $1. Calculate the financial expectation for this game. 5 A medical test for a disease does not always give the correct result. A positive test indicates that the patient has the disease. The two-way table below shows the results of a new screening test for the disease. It was tested on a group of people, some of whom were known to be suffering from the disease, some of whom were not. Test results Accurate

Not accurate

Total

28

2

30

Without disease

164

6

170

Total

192

8

With disease

CHAPTER

test yourself

7

a b c d e

How many people were tested for the disease? What percentage of the results were accurate? How many patients tested positive to the disease? What percentage of patients with the disease were correctly diagnosed by the new test? Based on the above results, what is the probability that a patient with the disease will have the disease detected by this test?

Annuities and loan repayments

8 syllabus reference Financial mathematics 5 • Annuities and loan repayments

In this chapter 8A Future value of an annuity 8B Present value of an annuity 8C Future and present value tables 8D Loan repayments

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

8.1

Finding values of n and r in financial formulas

8.2

Calculating simple interest

1 Find the value of n and r in for each of the following investments. a Interest of 8% p.a. for 5 years, with interest calculated annually b Interest of 6% p.a. for 4 years, with interest calculated six-monthly c Interest of 7.6% p.a. for 3 years, with interest calculated quarterly d Interest of 9.6% p.a. for 10 years, with interest calculated monthly e Interest of 24% p.a. for November, with interest calculated daily

2 Find the simple interest on each of the following investments. a $25 000 invested at 5% p.a. for 4 years b $15 500 invested at 8.2% p.a. for 6 years c $42 000 invested at 9.4% p.a. for 18 months

8.3

Calculating compound interest

3 Find the compound interest earned on each of the following investments. a $12 000 invested at 6% p.a. for 3 years, with interest compounded annually b $35 000 invested at 8% p.a. for 5 years, with interest compounded six-monthly c $56 000 invested at 7.2% p.a. for 4 years, with interest compounded quarterly

8.4

Reading financial tables

4 The table below shows the amount to which $1 will grow under compound interest. Interest rate per period Periods

6%

7%

8%

9%

1

1.060

1.070

1.080

1.090

2

1.123

1.145

1.166

1.188

3

1.191

1.225

1.260

1.295

4

1.262

1.311

1.360

1.412

Use the table to find the future value of each of the following investments. a $8000 at 6% for 2 years, with interest compounded annually b $12 500 at 8% p.a. for 3 years, with interest compounded annually c $18 000 at 12% p.a. for 2 years, with interest compounded six-monthly

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239

Future value of an annuity An annuity is a form of investment involving regular periodic contributions to an account. On such an investment, interest compounds at the end of each period and the next contribution to the account is then made. Superannuation is a common example of an annuity. Here, people invest in a fund on a regular basis, the interest on the investment compounds, while the principal is added to for each period. The annuity is usually set aside for a person’s entire working life and is used to fund retirement. It may also be used to fund a long-term goal, such as a trip in 10 years’ time. To understand the growth of an annuity, we need to revise compound interest. The compound interest formula is: A = P(1 + r)n where A is the final balance, P is the initial quantity, r is the interest rate per compounding period and n is the number of compounding periods.

WORKED Example 1

Calculate the value of a $5000 investment made at 8% p.a. for 4 years. THINK 1 2 3 4

WRITE

Write the values of P, r and n. Write the formula. Substitute values for P, r and n. Calculate the value of A.

P = $5000, r = 0.08, n = 4 A = P(1 + r)n A = $5000 (1.08)4 A = $6802.44

An annuity takes the form of a sum of compound interest investments. Consider the case of a person who invests $1000 at 10% p.a. at the end of each year for five years. To calculate this, we would need to calculate the value of the first $1000 that is invested for four years, the second $1000 that is invested for three years, the third $1000 that is invested for two years, the fourth $1000 that is invested for one year and the last $1000 that is added to the investment.

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WORKED Example 2 Calculate the value of an annuity in which $1000 is invested at the end of each year at 10% p.a. for 5 years. THINK 1 Use the compound interest formula to calculate the amount to which the first $1000 will grow. 2 Use the compound interest formula to calculate the amount to which the second $1000 will grow. 3 Use the compound interest formula to calculate the amount to which the third $1000 will grow. Use the compound interest formula to 4 calculate the amount to which the fourth $1000 will grow. 5 Find the total of the separate $1000 investments, remembering to add the final $1000.

WRITE A = P(1 + r)n A = $1000 1.14 A = $1464.10 A = P(1 + r)n A = $1000 1.13 A = $1331.00 A = P(1 + r)n A = $1000 1.12 A = $1210.00 A = P(1 + r)n A = $1000 1.1 A = $1100.00 Total value = $1464.10 + $1331.00 + $1210.00 Total value = + $1100.00 + $1000 Total value = $6105.10

In most cases it is more practical to calculate the total value of an annuity using a formula. The amount to which an annuity grows is called the future value of an annuity and can be calculated using the formula: ( 1 + r )n – 1 A = M ---------------------------r where M is the contribution per period paid at the end of the period, r is the interest rate per period expressed as a decimal, and n is the number of deposits. ( 1 + r )n – 1 For the above example: A = M ---------------------------r

1.1 5 – 1 = $1000 -----------------0.1

= $6105.10

WORKED Example 3 Bernie invests $2000 in a retirement fund at 5% p.a. interest compounded annually at the end of each year for 20 years. Calculate the future value of this annuity at retirement. THINK 1 Write the values of M, r, and n.

WRITE M = $2000, r = 0.05, n = 20

2

Write the formula.

( 1 + r )n – 1 A = M ---------------------------r

3

Substitute values for M, r and n.

1.05 20 – 1 A = $2000 -----------------------0.05

4

Calculate.

A = $66 131.91

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241

In some examples, calculations will need to be made when contributions are made more often than once a year and when interest compounds more often than once a year.

WORKED Example 4 Christina invests $500 in a fund every 6 months at 9% p.a. interest, compounding six-monthly for 10 years. Calculate the future value of the annuity after 10 years. THINK

WRITE 9% p.a. = 4.5% for 6 months So, r = 0.045 and n = 20.

1

Write the values of M, r and n by considering the interest rate as 4.5% per interest period and 20 interest periods.

2

Write the formula.

( 1 + r )n – 1 A = M ---------------------------r

3

Substitute for M, r and n.

1.045 20 – 1 A = $500 --------------------------0.045

4

Calculate.

A = $15 685.71

If we rearrange the formula for an annuity to make M (the contribution per period) the subject of the formula, we have: Ar M = --------------------------( 1 + r )n – 1 This formula would be used when we know the final amount to be saved and wish to calculate the amount of each regular deposit.

WORKED Example 5 Vikki has the goal of saving $10 000 in the next five years. The best interest rate that she can obtain is 8% p.a., with interest compounded annually. Calculate the amount of each annual contribution that Vikki must make. THINK

WRITE

1

Write the values of A, r and n.

A = $10 000, r = 0.08, n = 5

2

Write the formula.

Ar M = --------------------------( 1 + r )n – 1

3

Substitute for A, r and n. Hint: insert brackets when using your calculator.

( 10 000 0.08 ) M = -------------------------------------( 1.08 5 – 1 )

4

Calculate the value of M.

M = $1704.56

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remember 1. The compound interest formula is: A = P(1 + r)n where A is the final balance, r is the interest rate per period expressed as a decimal and n is the number of compounding periods. 2. An annuity is a form of investment where periodical equal contributions are made to an account, with interest compounding at the end of each period. 3. The value of an annuity is calculated by adding the value of each amount contributed as a separate compound interest investment. 4. We can calculate the value of an annuity by using the formula: ( 1 + r )n – 1 A = M ---------------------------r where M is the contribution per period, paid at the end of the period, r is the interest rate per period expressed as a decimal and n is the number of deposits. 5. The amount of each contribution to annuity to reach a certain goal can be calculated using the formula: Ar M = --------------------------( 1 + r )n – 1

SkillS

8A HEET

8.1

WORKED

Example

1

SkillS

Finding values of n and r in financial formulas HEET

8.2 Calculating simple interest WORKED

SkillS

Example

HEET

8.3 Calculating compound interest

2

Future value of an annuity

1 Calculate the value after 5 years of an investment of $4000 at 12% p.a., with interest compounded annually. 2 Calculate the value to which each of the following compound interest investments will grow. a $5000 at 6% p.a. for 5 years, with interest calculated annually b $12 000 at 12% p.a. for 3 years, with interest calculated annually c $4500 at 8% p.a. for 4 years, with interest compounded six-monthly d $3000 at 9.6% p.a. for 3 years, with interest compounded six-monthly e $15 000 at 8.4% p.a. for 2 years, with interest compounded quarterly f $2950 at 6% p.a. for 3 years, with interest compounded monthly 3 At the end of each year for four years Rodney invests $1000 in an investment fund that pays 7.5% p.a. interest, compounded annually. By calculating each investment of $1000 separately, use the compound interest formula to calculate the future value of Rodney’s investment after four years. 4 Caitlin is saving for a holiday in two years and so every six months she invests $2000 in an account that pays 7% p.a. interest, with the interest compounding every six months. a Use the compound interest formula to calculate the amount to which the: i first investment of $2000 will grow ii second investment of $2000 will grow iii third investment of $2000 will grow iv fourth investment of $2000 will grow. b If Caitlin then adds a final deposit of $2000 to her account immediately before her holiday, what is the total value of her annuity?

Chapter 8 Annuities and loan repayments

WORKED

Example

( 1 + r )n – 1 5 Use the formula A = M ---------------------------r

243

to find the future value of an annuity in

3

which $1000 is invested each year for 25 years at an interest rate of 8% p.a. 6 When baby Shannon was born, her grandparents deposited $500 in an account that pays 6% p.a. interest, compounded annually. They added $500 to the account each birthday, making the last deposit on Shannon’s 21st birthday. a How many deposits of $500 were made? b The investment was given to Shannon as a 21st birthday present. What was the total value of the investment at this point? (Hint: Use the answer to part a.) c Shannon’s grandparents advised Shannon to keep adding $500 to the investment each birthday so that she had a retirement fund at age 60. If Shannon follows this advice, what will the investment be worth at age 60? (Assume Shannon makes the last deposit on her 60th birthday.) 7 Calculate the future value of each of the following annuities. a $2000 invested at the end of each year for 10 years, at an interest rate of 5% p.a. b $5000 invested at the end of each year for 5 years, at an interest rate of 8% p.a. c $10 000 invested at the end of each year for 20 years, at an interest rate of 7.5% p.a. d $500 invested at the end of each year for 30 years, at an interest rate of 15% p.a. e $25 000 invested at the end of each year for 4 years, at an interest rate of 9.2% p.a. 8 Darlene is saving for a deposit on a unit. She hopes to buy one in four years and needs a $30 000 deposit, so she invests $5000 per year in an annuity at 7.5% p.a. starting on 1 January 2007. a After the last deposit is made on 1 January 2011, how many deposits has Darlene made? b Use the annuity formula to calculate if Darlene would have saved enough for her deposit. c How much interest was paid to Darlene on this annuity? WORKED

Example

4

9 At the end of every six months Jason invests $800 in a retirement fund which pays interest at 6% p.a., with interest compounded six-monthly. Jason does this for 25 years. Calculate the future value of Jason’s annuity after 25 years. 10 Calculate the future value of each of the following annuities on maturity. a $400 invested at the end of every six months for 12 years at 12% p.a., with interest compounded six-monthly b $1000 invested at the end of every quarter for 5 years at 8% p.a., with interest compounded every quarter c $2500 invested at the end of each quarter at 7.2% p.a., for 4 years with interest compounded quarterly d $1000 invested at the end of every month for 5 years at 6% p.a., with interest compounded monthly 11 multiple choice The interest earned on $10 000 invested at 8% p.a. for 10 years, with interest compounded annually, is: A $11 589.25 B $21 589.25 C $134 865.62 D $144 865.62

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12 multiple choice Tracey invests $500 in a fund at the end of each year for 20 years. The fund pays 12% p.a. interest, compounded annually. The total amount of interest that Tracey earns on this fund investment is: A $4323.15 B $4823.23 C $26 026.22 D $36 026.22 13 Thomas has the goal of saving $400 000 for his retirement in 25 years. If the best interest rate that Thomas can obtain is 10% p.a., with interest compounded annually, 5 calculate the amount of each annual contribution that Thomas will need to make.

WORKED

Example

14 Calculate the amount of each annual contribution needed to obtain each of the following amounts. a $25 000 in 5 years at 5% p.a., with interest compounded annually b $100 000 in 10 years at 7.5% p.a., with interest compounded annually c $500 000 in 40 years at 8% p.a., with interest compounded annually 15 Leanne is 24 years old and invests $30 per week in her superannuation fund. Leanne’s employer matches this amount. a If Leanne plans to retire at 60, calculate the total that Leanne will contribute to the fund at this rate. b Calculate the total contributions that will be made to the fund at this rate. c If the fund returns an average 4% p.a. interest, compounded annually, calculate the future value of Leanne’s superannuation.

Computer Application 1 Annuity calculator EXCE

et

reads L Sp he

Annuity calculator

Access the spreadsheet ‘Annuity calculator’ from the Maths Quest General Mathematics HSC Course CD-ROM. The spreadsheet will show you the growth of an annuity in which $1000 is invested at the end of each year for 20 years at a rate of 8% p.a. interest, compounding annually.

Chapter 8 Annuities and loan repayments

245

1. The spreadsheet shows that after 20 years the value of this investment is $45 761.96. Below is the growth of the annuity after each deposit is made. This will allow you to see the growth for up to 30 deposits. From the Edit menu, use the Fill Down functions on the spreadsheet to see further. 2. Click on the tab, ‘Chart1’. This is a line graph that shows the growth of the annuity for up to 30 deposits. 3. Change the size of the deposit to $500 and the compounding periods to 2. This will show how much benefit can be achieved by reducing the compounding period. 4. Check your answers to the previous exercise by using the spreadsheet.

1 1 Find the future value of $5000 invested at 10% p.a. for 6 years, with interest compounded annually. 2 Find the total amount of interest earned on an investment of $3200 invested for 4 years at 8% p.a., with interest compounded every six months. 3 Find the future value of an annuity of $1600 invested every year for 5 years at 12% p.a., with interest compounded annually. 4 Find the future value of an annuity of $2000 invested every year for 30 years at 7.2% p.a., with interest compounded annually. 5 Find the future value of an annuity in which $400 is invested every three months for 12 years at 8% p.a., with interest compounded quarterly. 6 Find the future value of an annuity in which $350 is invested each month for 10 years at 9.2% p.a. interest, compounding every six months. 7 Find the interest earned on an annuity of $750 invested per year for 10 years at 8.5% p.a., with interest compounding annually. 8 Find the amount of each annual contribution needed to achieve a future value of $100 000 if the investment is made for 10 years at an interest rate of 11% p.a., with interest compounding annually. 9 Find the amount of each quarterly contribution needed to save $15 000 in five years at 12% p.a., with interest compounding quarterly. 10 Find the amount of each six-monthly contribution to an annuity if the savings goal is $50 000 in 15 years and the interest rate is 8% p.a., with interest compounding sixmonthly.

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Present value of an annuity To compare an annuity with a single sum investment, we need to use the present value of the annuity. The present value of an annuity is the single sum of money that, invested on the same terms as the annuity, will produce the same financial result. To calculate the present value of an annuity, N, we can use the formula: A N = ------------------n(1 + r ) where A is the future value of the annuity r is the percentage interest rate per compounding period, expressed as a decimal n is the number of deposits to be made in the annuity

WORKED Example 6 Ashan has an annuity that has a future value of $500 000 on his retirement in 23 years. The annuity is invested at 8% p.a., with interest compounded annually. Calculate the present value of Ashan’s annuity. THINK

WRITE

1

Write the values of A, r and n.

A = $500 000, r = 1.08, n = 23

2

Write the formula.

A N = ------------------n(1 + r )

3

Substitute for A, r and n.

4

Calculate.

500 000 N = -----------------1.08 23 N = $85 157.64

In many cases you will not know the future value of the annuity when calculating the present value. You will know only the amount of each contribution, M. We know that: A N = ------------------n(1 + r ) ( 1 + r )n – 1 Using the formula A = M ---------------------------r

to substitute for A gives:

( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n This formula allows us to calculate the single sum needed to be invested to give the same financial result as an annuity where we are given the size of each contribution.

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247

WORKED Example 7 Jenny has an annuity to which she contributes $1000 per year at 6% p.a. interest, compounded annually. The annuity will mature in 25 years. Calculate the present value of the annuity. THINK

WRITE

1

Write the values of M, r and n.

M = $1000, r = 0.06, n = 25

2

Write the formula.

( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n

3

Substitute for M, r and n.

N = 1000

4

Calculate.

N = $12 783.36

1.06 25 – 1 ------------------------------0.06 1.06 25

This present value formula can be used to compare investments of different types. The investment with the greater present value will produce the greater financial outcome over time.

WORKED Example 8 Which of the following investments would give the greater financial return? Investment A: an annuity of $100 deposited per month for 20 years at 12% p.a. interest, compounding six-monthly Investment B: a single deposit of $10 000 invested for 20 years at 12% p.a., with interest compounding six-monthly THINK

WRITE

3

The investments can be compared by calculating the present value of the annuity. Consider the deposits of $100 per month to be $600 every six months. Write the values of M, r and n.

4

Write the formula.

( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n

5

Substitute for M, r and n.

N = $600

6

Calculate. Make a conclusion.

N = $9027.78 The annuity has a lower present value than the single investment. Therefore, the investment of $10 000 will produce a greater outcome over 20 years.

1

2

7

M = $600, r = 0.06, n = 40

1.06 40 – 1 ------------------------------0.06 1.06 40

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remember 1. The present value of an annuity is the single sum that can be invested under the same terms as an annuity and will produce the same financial outcome. 2. The present value of an annuity can be calculated using the formula: A N = ------------------n(1 + r ) when we know the future value of the annuity. 3. If we know the amount of each contribution of the annuity, we can calculate the present value using the formula ( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n where M is the contribution per period, paid at the end of the period r is the percentage interest rate per compounding period (expressed as a decimal) n is the number of interest periods 4. Investments can be compared using the present value formula. The investment with the greater present value will produce the greater financial outcome over time.

8B WORKED

Example

6

Present value of an annuity

1 Calculate the present value of an investment that is needed to have a future value of $100 000 in 30 years’ time if it is invested at 9% p.a., with interest compounded annually. 2 Calculate the present value of an investment required to generate a future value of: a $20 000 in 5 years’ time at 10% p.a., with interest compounded annually b $5000 in 4 years’ time at 7.2% p.a., with interest compounded annually c $250 000 in 20 years’ time at 5% p.a., with interest compounded annually. 3 Calculate the present value of an investment at 7.2% p.a., with interest compounded quarterly, if it is to have a future value of $100 000 in 10 years’ time. 4 Calculate the present value of the investment required to produce a future value of $500 000 in 30 years’ time at 9% p.a., with interest compounded: a annually b six-monthly c quarterly d monthly

WORKED

Example

7

5 Craig is paying into an annuity an amount of $500 per year. The annuity is to run for 10 years and interest is paid at 7% p.a., with interest compounded annually. Calculate the present value of this annuity. 6 Calculate the present values of each of the following annuities. a $1000 per year for 30 years at 8% p.a., with interest compounded annually b $600 per year for 20 years at 7.5% p.a., with interest compounded annually c $4000 per year for 5 years at 11% p.a., with interest compounded annually d $200 per month for 25 years at 8.4% p.a., with interest compounded annually

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249

7 Darren pays $250 per month into an annuity that pays 5.6% p.a. interest, compounded quarterly. If the annuity is to run for 10 years, calculate the present value of the annuity. 8 Calculate the present value of a 40-year annuity with interest at 9.6% p.a., compounded monthly, if the monthly contribution to the annuity is $50. 9 multiple choice An annuity is at 12% p.a. for 10 years, with interest compounded six-monthly, and has a future value of $100 000. The present value of the annuity is: A $31 180.47 B $32 197.32 C $310 584.82 D $320 713.55 10 multiple choice An annuity consists of quarterly deposits of $200 that are invested at 8% p.a., with interest compounded quarterly. The annuity will mature in 23 years. The present value of the annuity is: A $1236.65 B $2074.21 C $8296.85 D $8382.72 11 Which of the following investments will have the greater financial outcome? Investment A: an annuity of $400 per year for 30 years at 6.9% p.a., with interest 8 compounded annually Investment B: a single investment of $5000 for 30 years at 6.9% p.a., with interest compounded annually

WORKED

Example

12 multiple choice Which of the following investments will have the greatest financial outcome? A An annuity of $1200 per year for 30 years at 8% p.a., with interest compounded annually B An annuity of $600 every six months for 30 years at 7.9% p.a., with interest compounded six-monthly C An annuity of $300 every quarter for 30 years at 7.8% p.a., with interest compounded quarterly D An annuity of $100 per month at 7.5% p.a., for 30 years with interest compounded monthly.

Work

13 Kylie wants to take a world trip in 5 years’ time. She estimates that she will need $25 000 for the trip. The best investment that Kylie can find pays 9.2% p.a. interest, compounded quarterly. a Calculate the present value of the investment needed to achieve this goal. b Kylie plans to save for the trip by depositing $100 per week into an annuity. Calculate if this will be enough for Kylie to achieve her savings goal (take 13 weeks = 1 quarter).

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Future and present value tables Problems associated with annuities can be simplified by creating a table that will show either the future value or present value of an annuity of $1 invested per interest period.

Computer Application 2 Future value of $1 Consider $1 is invested into an annuity each interest period. The table we are going to construct on a spreadsheet shows the future value of that $1. 1. Open a new spreadsheet. 2. Type in the following information as shown in step 3. 3. In cell B4 enter the formula =((1+B$3)^$A4-1)/B$3. (This is the future value formula from exercise 2A with the value of M omitted, as it is equal to 1.) Format the cell, correct to 4 decimal places. 4. Highlight the range of cells B3 to M13. From the Edit menu, use Fill Down and Fill Right functions to copy the formula to all other cells in this range.

This completes the table. The table shows the future value of an annuity of $1 invested for up to 10 interest periods at up to 10% per interest period. You can extend the spreadsheet further for other interest rates and longer investment periods. The following table is the set of future values of $1 invested into an annuity. This is the table you should have obtained in computer application 2. A table such as this can be used to find the value of an annuity by multiplying the amount of the annuity by the future value of $1.

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Future values of $1 Interest rate (per period) Period

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

1

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2

2.0100 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 2.1100 2.1200

3

3.0301 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100 3.3421 3.3744

4

4.0604 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410 4.7097 4.7793

5

5.1010 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6.2278 6.3528

6

6.1520 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7.9129 8.1152

7

7.2135 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 9.7833 10.0890

8

8.2857 8.5380 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 11.8594 12.2997

9

9.3685 9.7546 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795 14.1640 14.7757

10

10.4622 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374 16.7220 17.5487

WORKED Example 9

Use the table to find the future value of an annuity into which $1500 is deposited at the end of each year at 7% p.a. interest, compounded annually for 9 years. THINK 1 2

WRITE

Look up the future value of $1 at 7% p.a. for 9 years. Multiply this value by 1500.

Future value = $1500

11.9780

Future value = $17 967

Just as we have a table for the future value of an annuity, we can create a table for the present value of an annuity.

Computer Application 3 Present value table The table we are about to make on a spreadsheet shows the present value of an annuity of $1 invested per interest period. 1. Open a new spreadsheet. 2. Enter the following information. 3. In cell B4 type the formula =((1+B$3)^$A4-1)/(B$3*(1+B$3)^$A4). 4. Drag from cell B4 to K13, and then from the Edit menu use the Fill Down and Fill Right functions to copy this formula to the remaining cells in your table.

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The table created in computer application 3 shows the present value of an annuity of $1 per interest period for up to 10% per interest period and for up to 10 interest periods. The table that you have generated is shown below. Present values of $1 Interest rate (per period) Period

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

1

0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 0.9009 0.8929

2

1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 1.7125 1.6901

3

2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 2.4437 2.4018

4

3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699 3.1024 3.0373

5

4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 3.6959 3.6048

6

5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 4.2305 4.1114

7

6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 4.7122 4.5638

8

7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 5.1461 4.9676

9

8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590 5.5370 5.3282

10

9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446 5.8892 5.6502

This table can be used in the same way as the future values table.

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WORKED Example 10 Liam invests $750 per year in an annuity at 6% per annum for 8 years, with interest compounded annually. Use the table to calculate the present value of Liam’s annuity. THINK 1 Use the table to find the present value of a $1 annuity at 6% for 8 interest periods. Multiply this value by 750. 2

WRITE

Present value = $750 6.2098 Present value = $4657.35

remember 1. A table of future values shows the future value of an annuity in which $1 is invested per interest period. 2. A table of present values shows the present value of an annuity in which $1 is invested per interest period. 3. A table of present or future values can be used to compare investments and determine which will give the greater financial return.

8C

Future and present value tables

1 Use the table of future values on page 251 to determine the future value of an annuity 8.4 of $800 invested per year for 5 years at 9% p.a., with interest compounded annually. 9 2 Use the table of future values to determine the future value of each of the following Reading financial annuities. tables a $400 invested per year for 3 years at 10% p.a., with interest compounded annually b $2250 invested per year for 8 years at 8% p.a., with interest compounded annually c $625 invested per year for 10 years at 4% p.a., with interest compounded annually d $7500 invested per year for 7 years at 6% p.a., with interest compounded annually 3 Samantha invests $500 every 6 months for 5 years in an annuity at 8% p.a., with interest compounded every 6 months. a What is the interest rate per interest period? b How many interest periods are there in Samantha’s annuity? c Use the table to calculate the future value of Samantha’s annuity. 4 Use the table to calculate the future value of each of the following annuities. a $400 invested every 6 months for 4 years at 14% p.a., with interest compounded sixmonthly b $600 invested every 3 months for 2 years at 12% p.a., with interest compounded quarterly c $100 invested every month for 5 years at 10% p.a., with interest compounded sixmonthly 5 Use the table of future values to determine whether an annuity at 5% p.a. for 6 years or an annuity at 6% p.a. for 5 years will produce the greatest financial outcome. Explain your answer.

WORKED

Example

SkillS

HEET

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6 multiple choice Use the table of future values to determine which of the following annuities will have the greatest financial outcome. A 6% p.a. for 8 years, with interest compounded annually B 8% p.a. for 6 years, with interest compounded annually C 7% p.a. for 7 years, with interest compounded annually D 10% p.a. for 5 years, with interest compounded six-monthly WORKED

Example

10

7 Use the table of present values on page 252 to determine the present value of an annuity of $1250 per year for 8 years invested at 9% p.a. 8 Use the table of present values to determine the present value of each of the following annuities. a $450 per year for 5 years at 7% p.a., with interest compounded annually b $2000 per year for 10 years at 10% p.a., with interest compounded annually c $850 per year for 6 years at 4% p.a., with interest compounded annually d $3000 per year for 8 years at 9% p.a., with interest compounded annually

2 1 Calculate the amount of interest earned on $10 000 invested for 10 years at 10% p.a., with interest compounding annually. 2 Calculate the future value of an annuity of $1000 invested every year for 10 years at 10% p.a., with interest compounding annually. 3 Calculate the future value of an annuity where $200 is invested each month for 5 years at 5% p.a., with interest compounding quarterly. 4 Calculate the amount of each annual contribution to an annuity that will have a future value of $15 000 if the investment is for 8 years at 7.5% p.a., with interest compounding annually. 5 Calculate the amount of each annual contribution to an annuity that will have a future value of $500 000 in 25 years when invested at 10% p.a., with interest compounding annually. 6 Calculate the present value of an annuity that will have a future value of $50 000 in 10 years at 10% p.a., with interest compounding annually. 7 Calculate the present value of an annuity that will have a future value of $1 000 000 in 40 years at 10% p.a., with interest compounding annually. 8 Calculate the present value of an annuity where annual contributions of $1000 are made at 10% p.a., with interest compounding annually for 20 years. 9 Use the table on page 251 to find the future value of $1 invested at 16% p.a. for 4 years, with interest compounding twice annually. 10 Use the answer to question 9 to calculate the future value of an annuity of $1250 every six months for 4 years, with interest of 16% p.a., compounding twice annually.

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Loan repayments When a loan is taken out and is repaid in equal monthly instalments, the pattern of repayments works similar to an annuity. Each month interest compounds on the balance owing on the loan and then a repayment is made. Consider a loan where the amount borrowed is equal to the present value of the annuity, N, and the amount paid on the loan each month is equal to the contribution to ( 1 + r )n – 1 - . the annuity per period, M. Use the formula for present value, N = M --------------------------r ( 1 + r )n To calculate the amount of each monthly repayment, we need to make M the subject of this formula. When we do this the formula becomes: r ( 1 + r )n M = N --------------------------( 1 + r )n – 1 In this formula, M is the amount of each repayment, N is the amount borrowed, r is the interest rate per repayment period as a decimal and n is the number of repayments to be made. This formula is not given to you on the formula sheet but will be given to you if it is needed to solve a problem in the exam.

WORKED Example 11 r( 1 + r )n Use the formula M = N --------------------------( 1 + r )n – 1

to calculate the monthly repayments on a loan of

$5000 to be repaid in monthly instalments over 4 years at an interest rate of 12% p.a. THINK

WRITE

1

Calculate the values of r and n.

r = 0.01 and n = 48

2

Write the formula.

r ( 1 + r )n M = N --------------------------( 1 + r )n – 1

3

Substitute for N, r and n.

M = 5000

4

Calculate.

M = $131.67

0.01 1.01 48 ------------------------------1.01 48 – 1

Having worked out the amount of each monthly repayment, we are also able to calculate the total cost of repaying a loan by multiplying the amount of each repayment by the number of repayments.

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WORKED Example 12 Calculate the total cost of repaying a $100 000 home loan at 9% p.a. in equal monthly repayments over a 25-year term. THINK

WRITE

1

Calculate the values of r and n.

r = 0.0075, n = 300

2

Write the formula.

r ( 1 + r )n M = N --------------------------( 1 + r )n – 1

3

Substitute for N, r and n.

M = 100 000

4

Calculate the amount of each monthly repayment. Calculate the total repayments on the loan.

M = $839.20 Total repayments = $839.20 300 Total repayments = $251 760

5

0.0075 1.0075 300 ---------------------------------------------1.0075 300 – 1

By increasing the amount of each repayment, we are able to shorten the term of the loan. There is no easy method to calculate the amount of time that it will take to repay a loan. To do this we use a ‘guess and refine’ method. We adjust the value of n in the formula until the amount of the repayment is reached.

WORKED Example 13 A $100 000 home loan is taken out over a 25-year term at an interest rate of 12% p.a. reducible interest. The minimum monthly repayment on the loan is $1053.22. How long will it take the loan to be repaid at $1200 per month? THINK 1

Calculate the value of r.

2

Write the formula.

3

Take a guess for the value of n (we will take 200 since for the original loan n = 300) and substitute.

WRITE r = 0.01 r ( 1 + r )n M = N --------------------------( 1 + r )n – 1 If n = 200, M = 100 000

4

5

Calculate the repayment with n = 200. As this is less than $1200 we need to further reduce the value of n. Substitute into the formula with n = 150.

= $1158.33 If n = 150, M = 100 000

6

Calculate the repayment. As the result is greater than $1200, we need to increase the value of n.

0.01 1.01 200 ---------------------------------1.01 200 – 1

= $1289.99

0.01 1.01 150 ---------------------------------1.01 150 – 1

Chapter 8 Annuities and loan repayments

THINK 7

WRITE

Substitute into the formula with n = 180.

If n = 180, M = 100 000

8 9

257

As this is approximately equal to $1200, it will take 180 months to repay the loan. Give a written answer.

0.01 1.01 180 ---------------------------------1.01 180 – 1

= $1200.17 It will take 15 years to repay the loan.

remember 1. By considering the amount borrowed in a loan as the present value of an annuity, we can use the present value formula to calculate the amount of each repayment. 2. The formula used to calculate the amount of each monthly repayment is: r ( 1 + r )n M = N --------------------------( 1 + r )n – 1 where N is the amount borrowed, r is the interest rate per period expressed as a decimal and n is the number of interest periods. 3. The total cost of a loan can be calculated by multiplying the amount of each repayment by the number of repayments to be made. 4. The length of time that it will take to repay a loan can be calculated by using guess and refine methods.

8D

Loan repayments

r ( 1 + r )n - . For questions 1 to 3 use the formula, M = N --------------------------( 1 + r )n – 1 1 Yiannis takes out a $10 000 loan over 5 years at 10% p.a. reducible interest with five equal annual repayments to be made. Use the formula to calculate the amount of each annual repayment. WORKED

Example

11

2 Use the formula to calculate the amount of each monthly repayment on a loan of $8000 to be repaid over 4 years at 12% p.a. 3 Use the formula to calculate the amount of each monthly repayment on each of the following loans. a $2000 at 12% p.a. over 2 years b $15 000 at 9% p.a. over 5 years c $120 000 at 6% p.a. over 20 years d $23 000 at 9.6% p.a. over 5 years e $210 000 at 7.2% p.a. over 25 years

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4 Javier and Diane take out a $175 000 home loan. If the interest rate on the loan is 8.4% p.a. reducible and the term of the loan is 25 years, calculate the amount of each monthly repayment. 5 Jiro purchases a computer on terms. The cash price of the computer is $3750. The terms are a deposit of 10% with the balance paid in equal monthly instalments at 9% p.a. reducible interest over 3 years. a Calculate Jiro’s deposit on the computer. b What is the balance owing on the computer? c Calculate the amount of each monthly repayment. 6 Jeremy and Patricia spend $15 000 on new furnishings for their home. They pay a 15% deposit on the furnishings with the balance paid in equal monthly instalments at 18% p.a. interest over 4 years. Calculate the amount of each monthly repayment. 7 Thanh is purchasing a car on terms. The cash price of the car is $35 000 and he pays a $7000 deposit. a What is the balance owing on the car? b If the car is to be repaid in equal weekly instalments over 5 years at an interest rate of 10.4% p.a. reducible interest, calculate the amount of each weekly payment. WORKED

Example

12

8 Ron borrows $13 500 to purchase a car. The loan is to be repaid in equal monthly instalments over a 3-year term at an interest rate of 15% p.a. Calculate the total repayments made on the loan. 9 Calculate the total repayments on each of the following loans. a $4000 at 8.4% p.a. reducible interest to be repaid over 2 years in equal monthly repayments b $20 000 at 13.2% p.a. reducible interest to be repaid over 6 years in equal monthly instalments c $60 000 at 7.2% p.a. reducible interest to be repaid over 15 years in equal monthly instalments d $150 000 at 10.8% p.a. reducible interest to be repaid over 20 years in equal monthly instalments 10 multiple choice A loan of $5000 is taken out at 9% p.a. reducible interest over 4 years. Which of the following will give the amount of each monthly repayment? A M = 5000

0.09 1.09 4 ----------------------------1.09 4 – 1

B M = 5000

0.09 1.09 48 ------------------------------1.09 48 – 1

C M = 5000

0.0075 1.0075 4 ----------------------------------------1.0075 4 – 1

D M = 5000

0.0075 1.0075 48 ------------------------------------------1.0075 48 – 1

11 multiple choice A loan of $12 000 is taken out at 12% p.a. reducible interest in equal monthly instalments over 5 years. The total amount of interest paid on the loan is: A $266.93 B $4015.80 C $7200 D $16 015.80

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12 A loan of $75 000 is taken out over 15 years at 9% p.a. reducible interest. The minimum monthly repayment is $760.70. Calculate how long it will take to repay the loan at 13 $1000 per month.

WORKED

13 A $150 000 loan is taken out over a 25-year term. The interest rate is 9.6% p.a. a Calculate the minimum monthly repayment. b Calculate the total repayments on the loan. c Calculate the length of time that it will take to repay the loan at $1600 per month. d Calculate the total saving on the loan by repaying the loan at $1600 per month.

Work

Example

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Types of loan arrangements Research one example of each of the following types of loans. A. Hire purchase agreement. This is the type of loan where a major item is purchased on terms. Usually a deposit is paid and then the balance plus interest is repaid over an agreed period of time. B. Personal loan This is a loan taken out from a bank or other financial institution. It can be used for any purpose and is unsecured. This means that there is no item of property that the bank can claim if repayments are not made. C. Home loan This is a secured loan, which means that, if the repayments are not made, the bank can claim the property and sell it to reclaim the amount outstanding on the loan. For each of the above loans, answer the following questions. 1 What is the interest rate? Is interest calculated at a flat or reducible rate? 2 Over what term can the loan be repaid? 3 How regularly must repayments be made? 4 Can additional repayments be made to shorten the term of the loan? 5 Can the interest rate be altered after repayments have begun to be made? 6 What other fees and charges apply to borrowing the money?

Most financial institutions will provide graphs that show the growth of an annuity and the declining balance of a loan. These graphs can be obtained by either visiting the bank or by going to the internet site for the relevant financial institution. Obtain a copy of a graph showing the growth of an investment and the declining balance of a loan. Alternatively, develop a spreadsheet that shows the growth of an annuity and the declining balance of a loan and use the charting function of the spreadsheet to draw the graph. Access the Word file ‘Annuities, Loans, Graphs’ from the Maths Quest General Mathematics HSC Course CD-ROM.

W

Computer Application 4 Graphs of annuities and loans ord

W

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summary Future value of an annuity • An annuity is where regular equal contributions are made to an investment. The interest on each contribution compounds as additions are made to the annuity. • The future value of an annuity is the value that the annuity will have at the end of a fixed period of time. • The future value of an annuity can be calculated using the formula: ( 1 + r )n – 1 A = M ---------------------------r where M is the contribution per period paid at the end of the period, r is the percentage interest rate per compounding period (expressed as a decimal) and n is the number of compounding periods. • The amount of each contribution per period in an annuity can be found using the Ar -. formula M = --------------------------( 1 + r )n – 1

Present value of an annuity • The present value of an annuity is the single sum that would need to be invested at the present time to give the same financial outcome at the end of the term. • The present value of an annuity can be calculated using the formula: A N = ------------------n(1 + r ) where A is the future value of the annuity. • An alternative formula to use is: ( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n where M is the contribution made to the annuity per interest period.

Use of tables • A table can be used to find the present or future value of an annuity. • The table shows the present or future value of $1 under an annuity. • The present or future value of $1 must be multiplied by the contribution per period to calculate its present or future value.

Loan repayments • The present value of an annuity formula can be used to calculate the amount of each periodical repayment in a reducing balance loan. This is done by considering the present value of an annuity as the amount borrowed and making M the subject of the formula. r ( 1 + r )n - . • The formula to be used is M = N --------------------------( 1 + r )n – 1 • The total amount to be repaid during a loan is calculated by multiplying the amount of each monthly repayment by the number of repayments to be made.

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CHAPTER review 1 Calculate the amount to which each of the following investments will grow. a $3500 at 12% p.a. for 3 years, with interest compounded annually b $2000 at 8% p.a. for 5 years, with interest compounded six-monthly c $15 000 at 9.2% p.a. for 8 years, with interest compounded quarterly d $4200 at 13.2% p.a. for 2 years, with interest compounded monthly

8A

2 $400 per year is invested into an annuity at 7% p.a., with interest compounded annually. Use

8A

( 1 + r )n – 1 the formula A = M ---------------------------r

to calculate the value of the annuity after 20 years.

( 1 + r )n – 1 3 Use the formula A = M ---------------------------r

to calculate the future value of each of the

8A

following annuities. a $500 invested per year for 25 years at 12% p.a., with interest compounded annually b $1000 invested every 6 months for 10 years at 9% p.a., with interest compounded six-monthly c $600 invested every 3 months for 5 years at 7.2% p.a., with interest compounded quarterly d $250 invested per month for 20 years at 12% p.a., with interest compounded monthly 4 An annuity consists of $100 deposits every month for 15 years. The interest rate is 9% p.a. and interest is compounded six-monthly. Find the future value of the annuity. Ar - to calculate the amount of each annual contribution to an 5 Use the formula M = --------------------------( 1 + r )n – 1 annuity to achieve a savings goal of $800 000 in 40 years at an interest rate of 8% p.a., with interest compounded annually. 6 Calculate the amount of each contribution to the following annuities. a $50 000 in 10 years at 6% p.a., with interest compounded annually and annual deposits b $250 000 in 30 years at 12% p.a., with interest compounded six-monthly and contributions made every six months c $120 000 in 20 years at 16% p.a., with interest compounding quarterly and contributions made quarterly A 7 Use the formula N = ------------------n- to calculate the present value of an annuity if it is to have a (1 + r ) future value of $350 000 in 30 years’ time at an interest rate of 10% p.a., with interest compounded annually. 8 Calculate the present value of the following annuities with a future value of: a $10 000 after 10 years at 5% p.a., with interest compounded annually b $400 000 after 40 years at 12% p.a., with interest compounded annually c $5000 after 5 years at 9% p.a., with interest compounded six-monthly d $120 000 after 8 years at 15% p.a., with interest compounded quarterly.

8A 8A 8A

8B 8B

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9 Phuong wants to purchase a car in 3 years. He feels that he will need $15 000. The best investment he can find is at 8.5% p.a., interest compounded quarterly. What is the present value of this investment? 10 Gayle invests $400 per year in an annuity. The investment is at 6% p.a., with interest compounded annually. Gayle plans to invest in the annuity for 25 years. Use the formula ( 1 + r )n – 1 N = M --------------------------r ( 1 + r )n

to calculate the present value of this annuity.

8B

11 When Joanne begins work at 18, she invests $100 per month in a retirement fund. The investment is at 9% p.a., with interest compounded six-monthly. a If Joanne is to retire at 60 years of age, what is the future value of her annuity? b What is the present value of this annuity?

8C

12 Use the table of future values of $1 on page 251 to calculate the future value of an annuity of $4000 deposited per year at 7% p.a. for 8 years, with interest compounded annually.

8C

13 Use the table of future values of $1 to calculate the future value of the following annuities. a $750 invested per year for 5 years at 8% p.a., with interest compounded annually b $3500 invested every six months for 4 years at 12% p.a., with interest compounded six-monthly c $200 invested every 3 months for 2 years at 16% p.a., with interest compounded quarterly d $1250 invested every month for 3 years at 10% p.a., with interest compounded six-monthly

8C

14 Use the table of present values of $1 on page 252 to calculate the present value of an annuity of $500 invested per year for 6 years at 9% p.a., with interest compounded annually.

8C

15 Use the table of present values to calculate the present value of each of the following annuities. a $400 invested per year for 5 years at 10% p.a., with interest compounded annually b $2000 invested every six months for 5 years at 14% p.a., with interest compounded six-monthly c $500 invested every three months for 2 1--- years at 16% p.a., with interest compounded 2 quarterly d $300 invested every month for 4 years at 12% p.a., with interest compounded half-yearly

8D

r ( 1 + r )n 16 Use the formula M = N --------------------------( 1 + r )n – 1

to calculate the amount of each monthly repayment

on a loan of $28 000 to be repaid over 6 years at 12% p.a.

8D

17 Scott borrows $22 000 to purchase a car. The loan is taken out over a 4-year term at an interest rate of 9.6% p.a., with the loan to be repaid in equal monthly repayments. a Calculate the amount of each monthly repayment. b Calculate the total amount that is repaid on the loan.

8D

18 Calculate the total repayments made on a home loan of $210 000 to be repaid in equal monthly repayments over 25 years at an interest rate of 8.4% p.a.

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19 Adam buys a new lounge suite for $4400 and pays for it on his credit card. The interest rate on the credit card is 21% p.a. Adam hopes to pay the credit card off in two years by making equal monthly repayments. a Calculate the amount of each monthly repayment that Adam should make. b Calculate the total amount that Adam will make in repayments. c Calculate the amount of interest that Adam will pay.

Practice examination questions 1 multiple choice Jenny invests $1000 per year for 20 years in an annuity. The interest rate is 6.5% p.a. and interest is compounded annually. The future value of the annuity is: A $3523.65 B $11 018.51 C $18 825.31 D $38 825.31 2 multiple choice Madeline invests $1000 per year for 20 years in an annuity. The interest rate is 6.5% p.a. and interest is compounded annually. The present value of the annuity is: A $3523.65 B $11 018.51 C $18 825.31 D $38 825.31 3 multiple choice Which of the following investments has the greatest future value after 10 years? A An annuity of $500 per year at 7.75% p.a., with interest compounded annually B An annuity of $250 per six months at 7.6% p.a., with interest compounded six-monthly C An annuity of $125 per quarter at 7.2% p.a., with interest compounded quarterly D A single investment of $3400 at 7.9% p.a., with interest compounded annually

8D

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4 multiple choice A loan of $80 000 is taken out over a 20-year term at an interest rate of 9% p.a. The monthly repayment is $719.78. What would the total saving be if the term were reduced to 15 years? A $91.63 B $16 493.40 C $21 991.20 D $26 693.40 5 Lien invests $2000 per year in an annuity. The term of the annuity is 20 years and the interest rate is 8% p.a., with interest compounding annually. a Calculate the future value of this annuity. b Calculate the present value of this annuity. c By how much will the future value of the annuity increase if Lien deposits $500 per quarter and interest is compounded quarterly? 6 Eddie has the goal of saving $1 000 000 over his working life, which he expects to be 40 years. Over the period of his working life, Eddie expects to be able to obtain an average 7% p.a. in interest with interest compounded every six months. a Calculate the present value of this annuity. Ar - to calculate the amount of each six-monthly b Use the formula M = --------------------------( 1 + r )n – 1 contribution to the annuity. c For the first 10 years of the annuity Eddie makes no contributions, preferring to direct all his money into paying off a mortgage. At that time he makes a single contribution to catch up on the annuity. What amount must Eddie deposit? 7 Jim and Catherine take out a $150 000 loan. The interest rate on the loan is 12% p.a. and the loan is to be repaid in equal monthly repayments over a 20-year term. r ( 1 + r )n a Use the formula M = N --------------------------( 1 + r )n – 1

CHAPTER

test yourself

8

to calculate the amount of each monthly

repayment. b Calculate the total amount of interest that Jim and Catherine will need pay on this loan. c Calculate the saving that Jim and Catherine will make by repaying the loan over a 12-year term.

Modelling linear and non-linear relationships

9 syllabus reference Algebraic modelling 4 • Modelling linear and non-linear relationships

In this chapter 9A 9B 9C 9D 9E

Linear functions Quadratic functions Other functions Variations Graphing physical phenomena

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

9.1

Substitution into a formula

9.2

Recognising linear functions

1 For each of the following linear equations, find the y-values corresponding to x, when x equals –3, –2, –1, 0, 1, 2 and 3. Show the results as a table of values. a y = 2x b y = 3x – 1 c y = 7 – 3x

2 State which of the following are linear functions.

9.3

a y = 4x – 1

b y = x2

d 2x – 3y + 5 = 0

e y = 2x

Gradient of a straight line

3 Calculate the gradient of the line joining the following points. a (1, 1) and (5, 6) b (4, 0) and (6, –6)

9.4

1 c y = --x f 2y = 5x

c (–3, 7) and (2, –3)

Graphing linear equations

4 Sketch the graph of the linear functions. a y = 3x b y = 2x – 3

c y = 5 – 2x

Chapter 9 Modelling linear and non-linear relationships

267

Linear functions As discussed in chapter 5, a linear function is a function in which the independent and dependent variables have only a power of 1. When graphed, these values form a straight line. An example of a linear function is y = 2x 1. The function is graphed by creating a table of values, plotting the pair of coordinates that are formed on a number plane and joining them with a straight line. The independent variable is x, and as such values of x are substituted into the equation to find the corresponding values of y. If we recognise the function as linear, we need to plot only three points. Two points are sufficient to fix a line and the third is a check. If all three points are not in a straight line, we know that an error has been made.

WORKED Example 1

Graph the equation y = 2x

1.

THINK 1

WRITE

Draw a table of values for x. (Choose three easy values of x.)

0

1

2

x

0

1

2

y

1

1

3

x y

2

3

Substitute each value of x into the equation to find the corresponding values of y. Plot each of the points formed on a number plane.

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

4

Join the points formed with a straight line and label the line with the equation.

1 2 3 4 5

x

y 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5

y = 2x – 1 1 2 3 4 5

x

The straight line in worked example 1 has the equation y = 2x – 1, which is written in gradient–intercept form. Any equation in the form y = mx + b is said to be in gradient–intercept form, because the gradient of the straight line is represented by m and the y-intercept is represented by b.

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This can be used to sketch any straight line. Considering worked example 1, we can begin by plotting the point (0, –1) as the y-intercept. Other points can then be plotted using the gradient, by plotting points 1 across and 2 up. That is, starting with (0, –1), we plot (1, 1), (2, 3), (3, 5) and so on. At this point it is worth remembering the gradient formula: vertical change in position m = --------------------------------------------------------------------horizontal change in position We use this formula when we know two points on the graph, and this is useful on many occasions to help us find the equation of a straight line.

Graphics Calculator tip! Graphing a linear function 1. From the MENU select GRAPH.

2. Delete any existing equation and enter Y1 = 2X – 1.

3. Press SHIFT F3 [V-Window]. This allows you to set the lower and upper limits to draw on both the xand y-axes. Enter the setting shown on the screen at right. 4. Press EXE to return to the previous screen, and then press F6 (DRAW) to draw the graph.

WORKED Example 2

A store owner finds that the number of televisions sold each week, N, decreases as the price, P, increases. This relationship can be given by the rule N = 200 0.2P. a Complete the table below. P

100

200

500

N b Graph the relationship between the number of televisions sold and their price. c How many televisions will be sold if they are priced at $900 each? d The store can sell only a maximum of 50 televisions each week. At what price should the televisions be sold?

Chapter 9 Modelling linear and non-linear relationships

THINK

WRITE

a Substitute the given values of P into the equation to find the corresponding values of N.

a

b

b

1

Plot the points given by the table. (Note: Only positive values of N and P are needed in this example.)

P

100

200

500

N

180

160

100

269

N 200 160 120 80 40 0

2

Join the points with a straight line and label the equation.

0 60 0 80 0 10 00

40

0

P

20

0

N 200 N = 200 – 0.2P

160 120 80 40 0

80 0 10 00

0 60

0 40

0

P

20

0

c Use the graph to find N when P = 900.

c When P = 900, N = 20; they will sell 20 televisions at $900 each.

d Use the graph to find P when N = 50.

d When N = 50, P = 750; the televisions should be sold for $750 each.

N 200 160

N = 200 – 0.2P

120 80 40

N = 0.05P

0 0

P

20 0 40 0 60 0 80 0 10 00

When two linear functions are graphed on the same pair of axes, the intersection of the two graphs shows the point where both equations hold true. This can have applications in a practical context. Consider worked example 2, in which the number of televisions sold each week was given by N = 200 0.2P. Now consider that the company producing the televisions is prepared to produce more if the price is higher. This is given by the rule N = 0.05P. When these two functions are graphed on the same pair of axes, we can see that the point of intersection is (800, 40).

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The company would want to sell all of the televisions that they produce and similarly would want to produce enough to meet this demand. This will be done if the televisions are sold at $800 each, as the company would be prepared to produce 40 per week at this price, and this would be the number that would be sold. Graphing linear functions can be used to determine profit, loss or break-even points. If cost and receipts are graphed, the difference between the y-values at any point will determine the profit or loss. The point where the graphs intersect will be the break-even point, where no profit or loss is made.

Graphics Calculator tip! Graphing a practical linear function Many practical functions need to have different limits than worked example 1, which we have drawn, with the simple limits of –5 to 5 on both the x- and y-axes. In worked example 2, negative values are not realistic, while the maximum possible value on the vertical axis is 200. We need a value on the horizontal axis that will show the greatest possible value of P. 1. From the MENU select GRAPH. 2. Delete any existing equation and enter Y1 = 200 – 0.2X. Note: We replace N with Y1 and P with X.

3. Press SHIFT F3 [V-Window]. This allows you to set the lower and upper limits to draw on both the xand y-axes. Enter the setting shown on the screen at right. 4. Press EXE to return to the previous screen, and then press F6 (DRAW) to draw the graph.

WORKED Example 3 The cost of producing shoes in Asia is given by the equation C = 2000 + 15n, where n is the number of pairs of shoes produced per day. The cost of producing shoes in Australia is given by the equation C = 1000 + 20n. a On the same pair of axes, graph the cost equations for producing shoes in Asia and Australia. b When is it more cost efficient to produce the shoes in Asia?

Chapter 9 Modelling linear and non-linear relationships

THINK

WRITE

a

a C = 2000 + 15n

1

Draw a table of values for each cost equation.

n

0

100

200

C

2000

3500

5000

271

C = 1000 + 20n

2

3

Plot a pair of points generated by each cost equation. Join each with a straight line labelling each with its equation.

n

0

100

200

C

1000

3000

5000

C 10 000

C = 1000 + 20n

8000 C = 2000 + 15n 6000 4000 2000 0

b It will be more efficient to produce the shoes in Asia after the point of intersection.

0

0 00 10

80

0

60

40

0

n

20

0

b If more than 200 pairs of shoes are produced per day, it will be cheaper to produce the shoes in Asia. This is because if n > 200 the value of C is less, if the shoes are produced in Asia.

the intersection of Graphics Calculator tip! Finding two graphs Consider the two graphs drawn in worked example 3. It will be cheaper to produce the shoes in Asia when the value of C = 200 + 15n is less than C = 1000 + 20n. To find when this occurs we need to locate the point of intersection of the two graphs as shown below. 1. From the MENU select GRAPH.

2. Delete any existing equations and enter Y1 = 2000 + 15X and Y2 = 1000 + 20X. Note that we replace C with Y1 and Y2 and n with X.

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Maths Quest General Mathematics HSC Course

3. Press SHIFT F3 [V-Window]. This allows you to set the lower and upper limits to draw on both the xand y-axes. Enter the setting shown on the screen at right. 4. Press EXE to return to the previous screen, and then press F6 (DRAW) to draw the graphs.

5. Press SHIFT F5 [G-Solv], followed by F5 [ISCT] (intersection). This will find the point of intersection and display the coordinates of this point. Be patient: this may take a moment. From this we can see that the intersection occurs at x = 200 and y = 5000. Interpreting this result in terms of the question shows us that when 200 pairs of shoes are produced the cost will be $5000 in either Australia or Asia. From that point on it will be cheaper to produce the shoes in Asia.

remember 1. A linear function has powers of only 1 for both the independent and dependent variables. 2. Linear functions, when graphed, will appear as a straight line and can be written in the form y = mx + b , where m is the gradient and b is the yintercept. 3. To graph a linear function, draw a table with at least three values for the independent variable and calculate the corresponding value for the dependent variable. Plot the pairs of coordinates generated and join with a straight line. 4. Linear functions can also be graphed using a graphics calculator. 5. Many practical situations can be graphed using linear functions. When two linear functions are graphed on the same pair of axes, the point of intersection will give some important information about the question.

SkillS

9A HEET

9.1

WORKED

Example

1

SkillS

Substitution into a formula HEET

9.2 Recognising linear functions

Linear functions

1 Graph the function y = x + 3. 2 Graph each of the following linear functions on separate axes. a y = 2x b y = 3x 2 c y= x d y = 5 2x e y = 1--- x + 3 f y=1 2

3 Consider the linear function 3x + 2y 6 = 0. a Copy and complete the table at right. b Graph the function 3x + 2y 6 = 0.

x y

0

1 --- x 4

2

4

Chapter 9 Modelling linear and non-linear relationships

WORKED

Example

4 The cost, C, of a taxi hire is given by the linear equation C = 3 + 1.5d, where d is the 9.3 distance travelled in kilometres. a Copy and complete the table below. Gradient of d

0

5

10

30

SkillS HEET

2

273 a straight line

9.4

C

HEET

b Graph the cost function for the taxi hire. c Use the graph to determine the cost of a 20 km taxi journey. d Katie has $24. How far can Katie afford to travel in a taxi?

SkillS

Graphing linear equations E

5 A concert promoter finds that the profit made on a performance is given by the Plotting equation P = 3n - 24 000, where n is the number of people who attend the concert. linear a Complete this table of values, and use it to graph the profit equation. graphs n P

0

10 000 0

b What profit will the promoter make if 20 000 people attend the concert? c What will be the financial outcome for the promoter if 5000 people attend the concert? d How many people will need to attend the concert for the promoter to break even? 6 It is found that the number of ice-creams that will be sold during a day at the beach decreases as the price of the ice-creams increases. The number of ice-creams that will be sold can be determined by the equation N = 1000 - 5P, where P is the price of the ice-creams in cents. a Graph the function. b How many ice-creams will be sold at $1 each? c If the ice-cream salesman has only 100 ice-creams to sell, at what price should he sell them? 7 Two linear functions are represented by y = 4 - x and y = 3x. a Graph both linear functions on the same pair of axes. b What is the point of intersection of the two graphs? 8 By graphing both functions on the same pair of axes, find the point of intersection of the graphs y = 2x - 6 and y = x - 1. 9 Find the point of intersection of the graphs x + 2y - 4 = 0 and y = 2x + 2. 10 A factory produces two types of computer games: game A and game B. a The factory can produce a maximum of 120 games per week. This can be repre3 sented by the linear equation A + B = 120. Graph this function. b Sales research shows that twice as many copies of game A will sell as game B. This can be represented by the equation 2A = B. On the same pair of axes graph this function. c Find the point of intersection of the two graphs and make a conclusion about the number of each game that should be produced by the factory each week.

WORKED

Example

sheet

L Spre XCE ad

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Maths Quest General Mathematics HSC Course

11 The cost of running an old refrigerator is $1.20 per day. This can be represented by the equation C = 1.2d. A new refrigerator will cost $900 but the cost to run will be only 30c per day. This can be represented by the equation C = 900 + 0.3d. a Copy and complete the table below. 0

d

1000

2000

C (old) C (new) b Graph both linear functions on the same pair of axes. c Find the point of intersection of the two graphs; hence, state after how many days it will become more economical to purchase a new refrigerator. 12 The cost, in dollars, of producing calculators can be given by the equation C = 15n + 1500, where n is the number of calculators produced. When selling the calculators the receipts can be given by the equation C = 20n. a Graph both linear functions on the same pair of axes. b Determine the number of calculators that need to be sold in order for the manufacturer to break even.

Conversion of temperature To convert a temperature from degrees Celsius to degrees Fahrenheit, you can use 9C the formula F = ------- + 32 . A simpler but less accurate way is to double degrees 5 Celsius and add 30. This approximation written as a formula becomes F = 2C + 30. 1 Use a spreadsheet or graphics calculator to graph each function on the same set of axes. 2 Describe the accuracy of the simpler formula and state the values for which it is most accurate.

Quadratic functions A quadratic function is a function that involves the independent variable (x) to the power of 2. The graph of a quadratic function is a parabola, a curved line that comes to either a minimum or maximum point. The graph of a quadratic function is again drawn by creating a table of values and plotting the pairs of coordinates generated. Because the graph is not a straight line, it is necessary to plot more than just three points to show the shape of the curve.

275

Chapter 9 Modelling linear and non-linear relationships

The most basic quadratic function is y = x2. The table of values is drawn showing at least nine values of x. x

3

2

1

y

9

4

1

1 --2

0

1 --2

1

2

3

1 --4

0

1 --4

1

4

9 y

Plotting these points gives the graph shown on the right. This graph has a minimum at (0, 0) and forms the basic shape for all parabolas. In general, the form of a quadratic function is y = ax2 + bx + c, and we need consider only positive values of x.

9 8 7 6 5 4 3 2 1 –4 –3 –2 –1 0 –1

WORKED Example 4

Consider the quadratic function y = x2 a Complete the table of values below. x y

0

1

2

3

b Graph the function for x 0. c State the minimum value of y = x2

4

x

5

4x + 7. WRITE

a Substitute each value of x into the function.

a

b

b

2

1 2 3 4

4x + 7.

THINK

1

y = x2

Plot the points generated by the table of values. Join the points plotted with a smooth curve.

x y

1 4

2 3

y 12 11 10 9 8 7 6 5 4 3 2 1 –1 0 –1

c The minimum value is the y-value at the point where the graph turns.

0 7

1 2 3 4 5

x

For y = x2 4x + 7, minimum value = 3.

3 4

4 7

5 12

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Maths Quest General Mathematics HSC Course

Graphics Calculator tip! Graphing a quadratic function Your graphics calculator can graph quadratic functions as well as linear functions (and many other types of functions as well). The calculator can also find any maximum or minimum point on the graph. Consider worked example 4. 1. From the MENU select GRAPH.

2. Delete any existing equation and enter Y1 = X2 – 4X + 7.

3. Press SHIFT F3 [V-Window]. In this course you will not need to consider negative values for x. Enter the setting shown on the screen at right.

4. Press EXE to return to the previous screen, and then press F6 (DRAW) to draw the graph.

5. Press SHIFT F5 [G-Solv], followed by F3 (MIN). This will find the minimum point and display the coordinates of that point. Be patient: this may take a moment. Note: 1. When setting the view window you do not have to get the limit right the first time. It may take a bit of trial and error, especially with the y-values to make sure that you have the minimum (or maximum) point in your display. 2. Any question that has a negative value of x2 (such as worked example 5) will be concave downwards and as such will have a maximum point and not a minimum point. In step 5 after pressing SHIFT F5 [G-Solv] you will need to press F2 (MAX). 3. The display on the graphics calculator can sometimes lead to a slight inaccuracy in the answer. This can be seen in step 6. In cases such as this we can see that the calculator should display X = 2. For quadratic functions that have a positive x2 term, the parabola is concave up. This means that the graph comes to a minimum point. When the x2 term is negative, the graph is concave down and the graph comes to a maximum point.

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Chapter 9 Modelling linear and non-linear relationships

WORKED Example 5 Graph the function y = 1 + 4x

x2.

THINK 1 2

3 4

WRITE

Draw a table of values. Substitute each value of x into the function. Plot the points formed by each pair of coordinates. Join the points with a smooth curve. Note: For this function, the maximum value is 5.

x

0

1

2

3

4

5

y

1

4

5

4

1

4

y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5

1 2 3 4 5

x

Quadratic models can be used to solve several practical situations.

WORKED Example 6 A ball is thrown in the air. Its height, h, after t seconds can be given by the formula h = 20t 5t 2. Graph the function to calculate the maximum height the ball will reach. THINK 1 2

3

4

Draw a table of values. Substitute the values of t to calculate the corresponding values of h. Plot the points formed by each pair of coordinates. Negative values of h can be ignored because height must be positive. Join the points formed with a smooth curve.

WRITE

t

0

1

2

3

4

5

h

0

15

20

15

0

25

h 20 16 12 8 4 –1 0 –1

5

The maximum height reached by the ball will be the h value at the turning point on the curve.

1 2 3 4 5

t

The maximum height reached by the ball is 20 m.

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Maths Quest General Mathematics HSC Course

remember 1. A quadratic function is a function where the independent variable is raised to the power of 2. 2. The graph of a quadratic function is a parabola. The parabola is a curved graph and can be drawn using a table of values that has several points to allow the shape of the graph to be formed. 3. If the x2 term is positive, the graph is concave up and has a minimum point. If the x2 term is negative, the graph is concave down and has a maximum point. 4. The maximum or minimum value in a practical situation is the dependent variable at the maximum or minimum point.

9B EXCE

et

reads L Sp he

WORKED

Example

4 Graphing quadratics

Quadratic functions

1 For the quadratic function y = x2 2x + 3: a copy and complete the table of values below x

0

1

2

3

4

5

y b draw the graph of the function c state the minimum value of x2 2x + 3. 2 For the quadratic function y = x2 4x 2, draw up a table of values and use the table to draw the graph of the function for x 0. 3 Graph each of the following functions for x a y = x2 6x + 5 b y = x2 + x + 5 c y = (x 2)2

0.

4 On the one set of axes, graph the following quadratic functions for x a y = x2 b y = 2x2 c y = 1--- x2

0.

2

5 On the one set of axes, graph each of the following quadratic functions for x a y = x2 b y = x2 + 2 c y = x2 3

0.

6 Use your answers to questions 5 and 6 to answer the following. a Describe the effect a coefficient of x2 has on the graph of a quadratic function. b Describe the effect adding a constant term has on the graph of a quadratic function. 7 Graph the function y = (x Explain why this occurs. WORKED

Example

5

1)2 + 4. Compare this with the graph of y = x2

8 Graph the function y = 2 + 2x

x2 for x

0.

9 Graph each of the following functions for x a y = 4 + 6x x2 b y = 8 x2

0. c y = (2

x)2

2x + 5.

Chapter 9 Modelling linear and non-linear relationships

279

10 multiple choice Which of the following functions is not a quadratic function? A y = x2 + 5x 4 B y = (x 4)2 x–2 C y = (x 2)(x + 2) D y = -----------x+2 11 multiple choice The graph drawn on the right could have the equation: A y = (x 2)2 + 3 B y = (x 2)2 3 2 C y = 4 (2 x) D y = (2 x)2 3 12 multiple choice Which of the following functions will produce the same graph as y = (x A y = x2 4x 1 B y = x2 4x + 19 2 C y = x 8x 1 D y = x2 8x + 19 13 Graph the quadratic function y = 2x2

4x + 8 for x

14 An object dropped from a height falls to Earth according to the equation 6 d = 5t 2, where d is the distance fallen in metres and t is the time in seconds since the object was dropped. a Draw the graph of d against t. b How far will the object fall in 4 seconds? c How long will it take for an object to fall a distance of 500 m?

WORKED

Example

15 The height of a ball which is thrown vertically upwards is given by the equation h = 30t 5t2. a Draw the graph of h against t. b Find the maximum height reached by the ball. c Find the length of time taken for the ball to return to Earth. 16 A rectangular field is to be made out of 100 m of fencing. If the length of the field is x metres: a show that the width of the field is (50 x) metres b show that the area is given by the quadratic function A = 50x x2 c draw the graph of the function d find the maximum area of the field and what dimension the field must be to give the maximum area.

0.

4)2 + 3?

Work

280

T SHEE

9.1

Maths Quest General Mathematics HSC Course

17 Another rectangular field is to be built with 100 m of fencing using a river as one side of the field as River shown on the right. x a Show that the area of the field can be given by the equation A = 100x 2x2. 100 – 2x b Graph the function. c Calculate the dimensions of the field so that the area of the field is a maximum.

Maximising areas 1 Sketch ten rectangles that each have a perimeter of 40 m. 2 Show the length, width and area of each rectangle in a table. 3 If the length of the rectangle is x: a explain why the width of the rectangle will be 20 x b write a quadratic equation for the area of the rectangle. 4 Use a spreadsheet or graphics calculator to graph your function. 5 What is the maximum area of the rectangle?

1 1 Graph the function y = 2x

3.

2 State the gradient of the function. 3 What is the y-intercept of the function? 4 State the linear function with a gradient of 3 and a y-intercept of 2. 5 Give an example of a linear function with a negative gradient. 6 Copy and complete the table below for the function y = 4 x

0

1

2

3

x2.

4

y x2 for x

0.

8 State the maximum value of the function y = 4

x2.

7 Draw the graph of the function y = 4 9 Is the function y = (x

5)2 concave up or concave down? Explain your answer.

10 Find the minimum value of y = (x

6)2 + 5.

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Chapter 9 Modelling linear and non-linear relationships

Other functions There are several other types of function that we should be able to graph. In each of these cases the graphs are curves and so several points should be found to demonstrate the shape of the curve.

Cubic functions A cubic function has the independent variable (x) raised to a power of 3.

WORKED Example 7 Graph the function y = 2x3. THINK 1 Draw a table of values. 2 Substitute values of x to find the corresponding values of y. 3 Plot the points generated by the table. 4 Join the points with a smooth curve.

WRITE x

0

1

2

3

y

0

2

16

54

y

y = 2x 3

100 80 60 40 20 0

x

1 2 3 4

Hyperbolas a A hyperbolic function is of the form y = --- , where a is a constant. For hyperbolas, x x 0, and so we graph only values of x > 0. As the value of x increases, the value of y will decrease, and therefore we need to look at values close to 0 when creating our table of values.

WORKED Example 8 2 Graph the function y = --- . x THINK 1 Draw a table of values. 2 Substitute the x-values into the equation to find the corresponding y-values. 3 4

Plot each pair of coordinates generated by the table. Join each point with a smooth curve.

WRITE x

1 --4

1 --2

1

2

3

4

y

8

4

2

1

2 --3

1 --2

y 9 8 7 6 5 4 3 2 1 0

y = 2–x

1 2 3 4

x

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Maths Quest General Mathematics HSC Course

Exponential graphs An exponential function is of the form y = ax or y = b(ax ), where a and b are both constants. An exponential graph can increase rapidly.

WORKED Example 9

Graph the function y = 2x. THINK

WRITE

1

Draw a table of values.

2

Substitute values of x to find the corresponding values of y.

3

Plot the points generated by the table.

4

Join the points with a smooth curve.

x

0

1

2

3

4

y

1

2

4

8

16

y 20 16 12 8 4 0

y = 2x 1 2 3 4 5

x

Graphics Calculator tip! Graphing other functions To graph the functions shown in worked examples 7, 8 and 9, enter the functions as shown below. In each case set the view window as shown in the diagrams on the x- and y-axes.

An exponential function of the form y = b(ax ) represents an example of exponential growth. These functions show the growth of an investment over a period of time. In examples where the value of a is between 0 and 1, the function models exponential decay. An example of this is the depreciation of an asset over time.

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WORKED Example 10 Glenn invests $10 000 at 8% p.a. with interest compounded annually. The growth of this investment can be given by the exponential function A = 10 000(1.08)n, where n is the number of years of the investment and A is the amount to which the investment grows. Graph the growth of this investment. THINK

WRITE

1

Draw a table of values.

2

Substitute values of n to find the corresponding values of A.

1

n

2

3

4

5

A 10 800 11 664 12 597 13 605 14 693 6

n

7

8

9

10

A 15 869 17 138 18 509 19 990 21 589 Plot the points generated by the table.

4

Join the points with a smooth curve.

35 000 Investment ($)

3

A = 10 000 (1.08)n

30 000 25 000 20 000 15 000 10 000 5000 0 0

5

10 15 Number of years

20

remember 1. A cubic equation is of the form y = ax3. a 2. A hyperbola is an equation of the form y = --- . In such a function x 0, and x we need to examine values of x close to 0 to observe the behaviour of the curve near the y-axis. 3. An exponential function is of the form y = b(ax ). An exponential function can be used to model a growth function such as the growth of an investment. If 0 < a < 1, the function will model an exponential decay such as the depreciation of an item.

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9C EXCE

et

reads L Sp he

WORKED

Example

7 Function grapher

Other functions

1 Graph the cubic function y = x3 for x

0.

2 Graph the following functions for x 0. a y = 3x3 b y = 1--- x3

c y = x3

2

WORKED

Example

8

WORKED

Example

9

4 3 Graph the hyperbolic function y = --- for x > 0. x 4 Graph each of the following functions for x > 0. 1 10 a y = --b y = -----x x

1 c y = – --x

5 Graph the function y = 3x. 6 Graph each of the following functions. a y = 4x b y = 10x

c y = ( 1--- )x 2

7 Graph the function y = 5(2x ). 8 multiple choice The equation of the graph shown on the right could be: A y = x3 B y = 3x C y = 3x 3 D y = --x

y

x

9 multiple choice 2 Which of the graphs shown below could be the graph of y = --- ? x A y B y C x D y

x

x

y

10 Ming Lai invests $1000 at 10% p.a. interest with interest compounded annually. This investment can be represented by the function A = 1000(1.1)n, where A is the amount 10 to which the investment grows and n is the number of years of the investment. Draw the graph of the function.

WORKED

Example

11 Kevin invests $50 000 at 12% p.a. interest, compounded annually. a Write an equation for the amount, A, to which the investment will grow in terms of the number of years of the investment, n. b Graph the function. c Use the graph to estimate the amount of time that it will take for the investment to reach $70 000. 12 A new car is purchased for $40 000. The car depreciates at the rate of 15% p.a. The value, V, of the car after a number of years, n, can be given by the equation V = 40 000(0.85)n. Graph this function.

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Chapter 9 Modelling linear and non-linear relationships

285

Compound interest The amount to which an investment will grow under compound interest can be found using the following formula: A = P(1 + r)n . Consider an investment of $10 000 at an interest rate of 8% p.a. 1 If interest is compounded annually, the amount to which the investment will grow can be given by the function A = 10 000(1.08)n, where n is the number of years. Graph this function using graphing software or a graphics calculator. 2 If interest is compounded six-monthly, the function becomes A = 10 000(1.04)2n. On the same set of axes graph this function. 3 Write a function that will show the amount to which the investment will grow if interest is compounded quarterly, and graph this function on the same set of axes. 4 Use the graphs drawn to describe the overall effect of shortening the compounding period.

Variations From our work on measurement we know that the area of a circle is given by the formula A = r 2, where A is the area and r is the radius of the circle. This is an example of a quantity (area) that varies in proportion with the power of another quantity (radius). This can be written as A r2. The symbol means in proportion to. In this example is the constant of variation, that is, the amount by which r 2 must be multiplied to calculate the area. An equation of the form y = ax2 or y = ax3 can be used to model several variations. In such cases we may need to calculate the constant of variation from some known or given information.

WORKED Example 11

It is known that y varies directly with the cube of x. It is known that y = 24 when x = 2. Write an equation connecting the variables x and y. THINK 1 2 3

4

WRITE

Write a proportion statement. Insert a constant of variation (k) to form an equation. Substitute the known values of x and y to find the value of k.

Replace the known value of k in the equation.

y x3 y = kx3 When x = 2, y = 24. 24 = k 23 = 8k k=3 y = 3x3

Once we have calculated the constant of variation, we are able to calculate one quantity given the other.

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WORKED Example 12 The surface area of a cube varies directly with the square of the length of the cube’s edge. a A cube of edge length 5.5 cm has a surface area of 181.5 cm2. Find the constant of variation. b Find the surface area of a cube with an edge length of 7.2 cm. THINK

WRITE

a

a s

1 2 3 4 5

b

1 2 3 4

Write a proportion statement choosing pronumerals s and e. Insert the constant of variation, k, to form an equation. Substitute known information.

s = ke2 When e = 5.5, s = 181.5 181.5 = k 5.52 181.5 = k 30.25 181.k = 6

Calculate 5.52. Solve the equation (divide by 30.25). Rewrite the proportion statement with k = 6. Substitute e = 7.2.

e2

b s = 6e2 When e = 7.2, s = 6 7.22 s = 311.04 The surface area of a cube with an edge of 7.2 cm is 311.04 cm2.

Calculate s. Give a written answer.

Hyperbolic functions represent inverse variations. These variations occur when one a quantity decreases as the other increases. An inverse variation is of the form y = --- . x

WORKED Example 13 It is known that y varies inversely with x and that when y = 8, x = 4. Write an equation connecting y with x. THINK 1

Write an inverse proportion statement.

2

Insert a constant of variation (k) to form an equation. Substitute the known values of x and y to find the value of k.

3

4

Replace the known value of k in the equation.

WRITE 1 --x k y = -x y

When x = 4, y = 8. k 8 = --4 k = 32 32 y = -----x

Chapter 9 Modelling linear and non-linear relationships

287

Such variations can also be applied to practical situations.

WORKED Example 14

It is known that the time taken for a journey varies inversely with speed. The trip takes 6 hours at 60 km/h. a Find the constant of variation. b How long will it take at 90 km/h? THINK

WRITE

a

a t

b

1

Write a proportion statement choosing pronumerals t and s.

2 3

Insert the constant of variation, k, to form an equation. Substitute known information.

4

Solve the equation (multiply by 60).

1

Rewrite the equation with k = 360.

2

Substitute s = 90.

3

Calculate t. Give a written answer.

4

1 --s

k t = -s When t = 6, s = 60 k 6 = -----60 k = 360 360 b t = --------s When s = 90, 360 t = --------90 t=4 The trip will take 4 hours at 90 km/h.

Graphics Calculator tip! Graphing variations Once we have established the value of the constant of variation, we can use the graphics calculator to graph the variation and then find the value of one variable given the other. Consider worked example 14. Having found in part a the value of k, we can 360 graph t = --------- . s 1. From the MENU select GRAPH.

2. Enter the equation Y1 = 360 ÷ X, which replaces t with Y1 and s with X.

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3. Press SHIFT F3 [V-Window]. In this situation you will not need to consider negative values for x. Enter the setting shown on the screen at right.

4. Press EXE to return to the previous screen, and then press F6 (DRAW) to draw the graph.

5. To find the value of t (Y) when s (X) = 20, press SHIFT F5 [G-Solv], then F6 ( �) for more options and then F1 (Y-CAL). Enter x = 90 and the calculator will return the y-value.

remember 1. A variation can be expressed as a function. 2. If one quantity varies as the square of another, the variation is of the form y = ax2. 3. If one quantity varies as the cube of another, the variation is of the form y = ax3. a 4. If one quantity varies inversely as another, the variation is of the form y = --- . x 5. An inverse variation occurs when one quantity decreases while the other decreases.

9D WORKED

Example

11

Variations

1 It is known that y varies directly with the square of x. If y = 88 when x = 4, write an equation connecting y with x. 2 It is known that b varies directly with the cube of a. When a = 6, b = 108. Write an equation connecting b with a. 3 It is known that the distance, d, an object will fall varies directly with the square of the time, t, it has been falling. An object that has been falling for 2 seconds falls a distance of 19.6 metres. a Write an equation connecting d with t. b Graph the relationship between d and t.

WORKED

Example

12

4 The surface area of a cube varies directly with the square of its side length. a A cube of side length 15 cm has a surface area of 1350 cm2. Find the constant of variation. b What is the surface area of a cube that has a side length of 6.2 cm?

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Chapter 9 Modelling linear and non-linear relationships

5 The area of a circle varies directly with the square of its radius. a If the area of a circle with side length 6 cm is 113.1 cm2, find the constant of variation. (Give your answer correct to 2 decimal places.) b What is the area of a circle with a radius of 12 cm? 6 The mass of an egg varies directly as the cube of the egg’s length. a An egg of length 5 cm has a mass of 31.25 g. Find the constant of variation. b What will be the mass of an egg with a length of 6 cm? c If an egg has a mass of 70 g, what would be the egg’s length? (Give your answer correct to 1 decimal place.) WORKED

Example

13

7 It is known that y varies inversely with x. When y = 10, x = 5; write an equation connecting y with x. 8 It is known that m varies inversely with n. When m = 0.5, n = 2; write an equation connecting m and n. 9 The time taken, t, to travel between two points varies inversely with the average speed, s, for the trip. If the journey takes 2.5 hours at 60 km/h: a write an equation that connects t with s b graph the relationship between t and s.

10 The time, t, taken to dig a trench varies inversely with the number of workers, n, digging. It takes four workers 5 hours to dig the trench. 14 a Find the constant of variation. b How long would it take 10 workers to dig the same trench?

WORKED

Example

11 The fuel economy, f, of a car varies inversely with the speed, s, at which it is driven. A car that averages 40 km/h has a fuel economy of 15 km/L. What will be the fuel economy of a car that averages 50 km/h? 12 In an electricity circuit, the current (measured in amps, a) is inversely proportional to the resistance (measured in ohms, r). When the resistance is 40 ohms, the current is measured at 3 amps. What will be the current when the resistance is 15 ohms?

Graphing physical phenonema In many cases, an algebraic function can be used to graph a physical situation. Consider the case of a sphere of radius r. The volume of a sphere can be given by the formula V = 4--- r 3. We can create a table of values that allows us to graph the function for 3 volume. r

0

1

2

V

0

4.19

33.51

3 113.10

4 268.08

5 523.60

10 4189

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We can then plot each pair of points from the table and join the points with a smooth curve. The graph shown at right shows the volume for a sphere of any radius.

4000 3500 3000 Volume

2500 2000 1500 1000

4 r3 V =— 3

500 0

5 10 15 Radius

WORKED Example 15 The surface area of a sphere is given by the formula A = 4 r2. a Complete the table below. r

0

1

2

3

4

5

6

7

8

9

10

A b Graph the surface area function. THINK

WRITE

a Substitute each value of r into the formula.

a

b

b

2

Plot each pair of points generated by the table. Join the points with a smooth curve and extrapolate the graph.

0

A

0 12.57 50.27 113.10 201.06 314.16 452.39 615.75 804.25 1017.88 1256.64

1

2

3

3000

4

5

6

7

8

9

10

A = 4π r2

2500 Area

1

r

2000 1500 1000 500 0

0

5

10 15 Radius

Many graphs have physical restrictions placed on them. Consider the case of a ball that is thrown vertically upwards. The height, h, of the ball at any time, t, can be given by the equation h = 15t 5t 2. The height of the ball must always be positive, and when the ball returns to Earth the graph stops as shown on the right.

h h = 15t – 5t 2

12 9 6 3 0

1

2

3

4

t

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Chapter 9 Modelling linear and non-linear relationships

When we graph several points, we try to estimate other values by interpolating (estimating values between given points by drawing the graph joining the points) or extrapolating (estimating values by extending the graph beyond the points given). Other graphs need to have restrictions placed upon them when we try to interpolate or extrapolate. There may be a limit placed upon one or both of the variables, and this will indicate a change in the graph.

WORKED Example 16 A cinema owner believes that more people will attend the movies on cold days and so believes the number of people attending each session of a movie varies inversely with the temperature of the day. When the temperature is 15°C, 80 people attend a movie. The cinema has a maximum of 120 seats, and the cinema owner believes that a minimum of 40 people will attend, regardless of temperature. a Write an equation connecting the number of people attending the movie, N, with the temperature, T. b Graph the relationship between attendance and temperature. THINK WRITE a 1 Write an inverse proportion 1 a N --statement. T k 2 Insert a constant of variation, k, to N = --form an equation. T When T = 15, N = 80. 3 Substitute the known values of N and k T to find the value of k. 80 = -----15 k = 1200 1200 4 Replace the known value of k in the N = -----------equation. T b

1 2 3

Create a table of values. Substitute each value of T into the equation. If the value of N > 120, then we enter 120 for N (maximum attendance); if N < 40, enter 40 for N (minimum attendance).

b

10 15

20

25

30

35

N 120 120 80

60

48

40

40

T

5

N 120 100 80 60 40 20

0

4

Plot the points and join with a smooth curve. The minimum and maximum attendance is drawn with a straight line.

10

20

30

40

T

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remember 1. An algebraic model can be used to represent many physical situations. 2. When modelling a situation, there may be restrictions on one or both of the variables.

9E

Graphing physical phenomena

1 The surface area of a cube is given by the formula A = 6s2. a Complete the table of values below. 15

WORKED

Example

s

0

1

2

3

4

5

A b Draw the graph to represent the surface area of a cube of a given side length. 2 The distance that an object will fall when dropped from a height can be given by the formula d = 5t 2, where d is in metres and t is in seconds. Draw a graph of the function. 3 A car is travelling at v km/h and the driver needs to brake. It takes 2.5 seconds to react and in that time the car will travel a distance of 0.7v m. The total stopping distance, d, can be given by the function d = 0.01v2 + 0.7v. a Copy and complete the table below. v

0

10

20

30

40

d b Draw the graph of the stopping distance of a vehicle. 4 Lorraine organises a lottery syndicate at her work. If they win a prize of $100 000, the amount is shared equally between the members of the syndicate. There must be at least 16 one member of the syndicate and a maximum of 10. a Write an equation putting the amount, A, each person receives in terms of the number of members, n. b Graph the function.

WORKED

Example

5 A car is purchased new for $40 000. After one year the depreciated value of the car is $30 000. After the first year the car depreciates at a rate of 20% p.a. a Copy and complete the table below. Age (years)

1

2

3

4

5

Value b The car will always be worth a minimum of $2000 in scrap metal and accessories. Graph the value of the car against the age of the car. 6 The mass of a newborn baby increases by 20% per month for the first four months of life. If the average mass of a newborn baby is 3.3 kg, graph the mass function up to n = 4.

Chapter 9 Modelling linear and non-linear relationships

7 A square piece of sheet metal has a side length of 12 m. A square of side length x m is to be cut from each corner of the sheet metal and the sides bent up to form an open rectangular prism. a What is the maximum possible value of x? b Show that the volume of the prism formed can be given by the function V = x(12 2x)2. c Graph the volume function.

293

12 m xm

8 The population of a city is growing at a rate of 5% p.a. If the population in 2000 is 1.5 million, the population function can be given by the function P = 1.5(1.05)n, where P is the population, in millions. The city cannot sustain a population greater than 4 000 000. a Complete the table below. Year

2007

2008

2009

2010

2011

b Plot the points given and extrapolate to graph the population function. c Use your graph to state when the population will reach its maximum sustainable level. d What will happen to the graph when it reaches this level?

Force of gravity When an object is dropped, the distance that it will fall in t seconds can be approximated by the formula d = 5t 2. The coefficient of t 2 is half the force of gravity (10 m/s2) and so will change if an object were to be dropped on another planet. For example, on the moon this equation would become d = 0.8t2. 1 Use a graphics calculator or graphing software to graph the equations for both the Earth and the moon. 2 Find out the force of gravity on other planets and compare the graphs formed with that for the Earth.

Work

Population (million) T SHEE

9.2

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summary Linear functions • Linear functions have powers of only 1 for both the independent and dependent variables and are graphed as straight lines. • To graph a linear function, a table of at least three values is drawn; the points generated are plotted on a number plane and then joined with a straight line. • The intersection of two linear functions will give the point where both conditions hold true.

Quadratic functions • A quadratic function is a function where the independent variable is raised to the power of 2. • The graph of a quadratic function is a parabola, a curved graph with either a minimum (positive x2 term) or a maximum (negative x2 term). • A quadratic function is graphed by plotting the points formed from a table of at least seven values.

Other functions • A cubic function uses a power of 3 for the independent variable. It is of the form y = ax3. a • A hyperbola is a function of the form y = --- . In a hyperbolic function, as one x variable increases the other decreases. • An exponential function is of the form y = ax. When a > 1, an exponential function models exponential growth, while if 0 < a < 1, the function models exponential decay. • Each of these functions is graphed by plotting points from a table of values.

Variations • A variation occurs when one quantity changes in proportion with another. • If one quantity varies directly with another, as one increases so does the other. • If the quantity varies directly with the square of the other, it can be expressed as a function in the form y = ax2. If it varies with the cube of another, it can be expressed in the form y = ax3. • An inverse variation occurs when one quantity decreases, while the other increases. a An inverse variation can be expressed in the form y = --- . x • The constant of variation, a, is calculated by using a known quantity of each variable. Once this has been calculated, if we know one quantity we can calculate the other.

Graphing physical phenomena • Algebraic models can be used to represent many physical situations. • When graphing physical phenomena, we need to consider any restrictions that may exist on one or both of the variables.

Chapter 9 Modelling linear and non-linear relationships

295

CHAPTER review 1 Graph each of the following linear functions. a y = 3x b y=x+3 d y = 5 3x e 2y = 4x 3

c y=2 x f 3x 2y + 6 = 0

2 The cost, C, of a taxi fare is given by the formula C = 3 + 0.4d, where d is the distance travelled by the car, in kilometres. a Copy and complete the table below. d

0

5

10

15

9A 9A

20

C b Graph the cost function. 3 At a fete, 400 cans of soft drink are purchased for $320. The cans are then sold for $1.25 each. a Write, as a linear function, an expression for the profit on the sale of the cans, where n is the number of cans sold. b Graph the profit function. c What will be the financial outcome if: i 300 cans are sold? ii 142 cans are sold? d How many cans will need to be sold for the drink stall to break even?

9A

4 Graph the linear functions y = 6

9A 9A

x and y = x + 2, and hence state the point of intersection.

5 Andrew needs to purchase a new washing machine. a A brand new washing machine will cost $1000, and running costs will be approximately 20c per wash. Express this as a linear function. b Alternatively, Andrew could purchase a second-hand washing machine for $200, but running costs will be about $1.00 per wash. Express this as a linear function. c Graph both linear functions on the same pair of axes. d By finding the point of intersection, find out after how many washes does it become more economical to purchase the new machine. 6 For the quadratic function y = x2 4x + 5: a copy and complete the table of values below x

0

1

2

3

4

9B 5

y b draw the graph of the function for x 0 c state the minimum value of the function y = x2 7 For the quadratic function y = x2 the graph for x 0.

2x

4x + 5.

2, draw a table of values and use the table to sketch

9B

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Maths Quest General Mathematics HSC Course

9B

8 Sketch each of the following quadratic functions for x a y = (x 4)2 b y = 5 x2

9B

9 An object is dropped from a height of 500 m. Its height above the ground at any time, t, is given by the function h = 500 5t 2. a Draw the graph of the function. b How many seconds does it take for the object to fall to Earth?

9B

10 A team of workers are digging a mine shaft. The number of kilograms of earth moved each hour by the team is given by the function E = 24n n2, where n is the number of workers digging the shaft. a Graph the function. b What is the maximum amount of earth that can be moved by the team of workers in one hour? How many workers are needed to move this amount of earth? c Explain possible reasons why the amount of earth moved each hour then begins to decrease as more workers are used.

9C

11 Graph each of the following cubic functions for x a y = x3 b y = 1--- x3

0. c y = 4 + 2x

x2

0.

2

9C

12 Graph each of the following hyperbolic functions for x > 0. 2 1 a y = --b y = --x x

9C

13 Graph each of the following exponential functions. a y = 2x b y = ( 1--- )x 2

9C

14 The average inflation rate is 4% p.a. In 2006 it cost the average family $500 per week in living expenses. The future cost of living, C, can be estimated using the function C = 500(1.04)n where n is the number of years since 2006. a Graph the cost of living function. b Use the graph to estimate the cost of living in 2016. c When will the cost of living first reach $1000 per week?

9C

15 If the value of a computer purchased for $5000 depreciates by 20% p.a., the future value of the computer, V, can be given by the equation V = 5000(0.8)n, where n is the age of the computer, in years. a Graph the function. b Find when the value of the computer is approximately $1000.

9D

16 It is known that y varies directly with the square of x. When x = 4, y = 80. Write an equation connecting x with y.

9D

17 The mass, m, of an egg varies directly with the cube of its length, l. An egg of length 5.5 cm, has a mass of 75 g. a Write an equation connecting m with l. b Find the mass of an egg with a length of 5 cm. c Find the length of a 50 g egg.

Chapter 9 Modelling linear and non-linear relationships

297

18 It is known that y varies inversely with x. When x = 8, y = 8; write an equation connecting y with x.

9D

19 The amount of food in a camp varies inversely with the number of people to feed. There is enough food to feed 100 campers for 10 days. a Write an equation connecting the amount of food, A, with the number of campers, n. b Calculate how long the food would last 125 campers. c If the food lasts for four days, calculate the number of campers.

9D

20 The area of a circle is given by the formula A = r2. a Complete the table of values below.

9E

r

0

1

2

3

4

5

A b Draw the graph of A against r. 21 A ball is thrown directly up in the air. The height, h, of the ball at any time, t, can be found using the equation h = 20t 5t 2. a Draw a graph of the height equation. b Use the graph to find: i the maximum height of the ball ii the time taken for the ball to fall back to earth.

9E

22 An investment of $10 000 at 6% p.a. can be modelled using the equation A = 10 000(1.06)n, where n is the number of years of the investment. a Graph the function. b Use your graph to estimate the value of the investment after 8 years. c Use your graph to find the amount of time that it will take for the investment to grow to $15 000.

9E

Practice examination questions 1 multiple choice Which of the following equations is not an example of a linear function? 2 A y = 2x + 1 B y = --C 2y = x + 1 D x + 2y + 1 = 0 x 2 multiple choice Which of the following quadratic equations is equivalent to y = (x 3)2 + 7? A y = x2 3x 2 B y = x2 3x + 16 2 C y = x 6x 2 D y = x2 6x + 16 3 multiple choice

y

The graph shown at right could have the equation: 2 A y = x2 B y = --x x 1 x Cy=2 D y = ( --- ) 2

x

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Maths Quest General Mathematics HSC Course

4 multiple choice It is known that y varies inversely with x. The variation can be modelled by the equation: A y = ax B y = ax2 a C y = ax3 D y = --x 5 As a fundraising activity, a school hires a cinema to show the premiere of a movie. The cost of hiring the cinema is $500. People are then charged $10 to attend the movie. a Write a function for the profit or loss made on the movie in terms of the number of people attending. b Graph the function. c Use the graph to calculate the number of people who must attend the movie for the school to break even. d A rival cinema offers to waive the hire fee but the school will receive only $5 per person attending. On the same axes graph the function P = 5n. e The school chose to pay the $500 and receive $10 per person. How many people must attend the premiere to make this the better of the two options? 6 A rock is thrown from a cliff 20 m above ground level. The height of the rock at any time is given by the quadratic function h = 20 + 15t 5t 2. a Copy and complete the table below. t

0

1

2

3

4

h b Graph the function and use your graph to find the maximum height reached by the ball. 2 7 a On the one set of coordinate axes, sketch the graphs of y = 2x3 and y = --- . x 2 b Use your graphs to find the point of intersection of the graphs y = 2x3 and y = --- . x

CHAPTER

test yourself

9

8 The growth of an investment made at 8% p.a. can be modelled by the equation y = 1.08x. a Graph the function. b Use your graph to determine the amount of time that it will take for the investment to double in value. c The depreciation of an item at 8% p.a. can be modelled by the equation y = 0.92x. Graph this function. d Use your graph to determine the amount of time that it will take for the item to halve in value.

Depreciation

10 syllabus reference Financial mathematics 6 • Depreciation

In this chapter 10A Modelling depreciation 10B Straight line depreciation 10C Declining balance method of depreciation 10D Depreciation tables

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

10.1

Graphing linear equations

1 Draw the graphs of the following equations. a y = 2x – 1 b y = 8 – 4x

10.2

Graphing exponential functions

2 Draw the graphs of the following equations for x 0. a y = 2x b y = (0.8)x

10.3

2

Solving linear equations

3 Solve the equations. a 7x – 5 = 79

10.4

c y = 5( 1--- )x

b 3000x – 500 = 12 500

c 6000 – 500x = 3500

Calculating compound interest

4 Calculate: a The amount to which $10 000 will grow at 6% p.a. over 5 years with interest compounded annually b The amount to which $50 000 will grow at 8.2% p.a. over 4 years with interest compounded six-monthly.

10.6

Reading financial tables

5 The table below shows the amount to which $1 will grow under compound interest. Interest rate per period Periods

6%

7%

8%

9%

1

1.060

1.070

1.080

1.090

2

1.123

1.145

1.166

1.188

3

1.191

1.225

1.260

1.295

4

1.262

1.311

1.360

1.412

Use the table to find: a the amount to which $10 000 will grow at 7% p.a. over 4 years with interest compounded annually. b the amount to which $50 000 will grow at 12% p.a. over 2 years with interest compounded sixmonthly.

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Modelling depreciation An asset is an item that has value to its owner. Many assets such as cars and computers lose value over time. This is called depreciation. Consider the case of a new motor vehicle. The value of the car depreciates the moment that you drive the car away from the showroom. This is because the motor vehicle is no longer new and if it were sold, it would have to be sold as a used car. The car then continues to lose value steadily each year.

Depreciation of motor vehicles Choose a make of car and find out the price for a new vehicle of this make and model. Look through NRMA’s Open Road magazine or the classified advertisements in the newspaper to find the price of the same model as a second-hand car. Age of car (years)

Price

New (0) 1 2 3 4 5 Draw a graph that shows the price of this car as it ages.

There are two types of depreciation: the straight line method and the declining balance method. The straight line method is where the asset depreciates by a constant amount each year. When this type of depreciation is graphed, a straight line occurs and the asset will reduce to a value of 0. In such a case, a linear function can be derived that will allow us to calculate the value of the item at any time. The function can be found using the gradient–intercept method. The purchase price of the asset (V0) will be the vertical intercept, and the gradient will be the negative of the amount that the item depreciates, D, each period. The equation of this linear function will be: V = V0

Dn

where V is the salvage value of the item and n is the age of the asset, in years. Note: Gradients for depreciation will always be negative.

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WORKED Example 1

The table below shows the declining value of a computer. Graph the value against time and write an equation for this function. Age (years)

Value ($)

New (0)

4000

1

3500

2

3000

3

2500

4

2000

5

1500

WRITE

1

Draw a set of axes with age on the horizontal axis and value on the vertical.

2

Plot each point given by the table.

3

Join all points to graph the function.

Value ($)

THINK

4500 4000 3500 3000 2500 2000 1500 1000 500 0 0123456789 Age (years)

4

Write the initial value as V0 and use the gradient to state D.

V0 = 4000, D = 500

5

Write the equation using V = V0

V = V0 Dt V = 4000 500t

Dt.

Note: To solve worked example 1 you can use the graphics calculator methods demonstrated in chapter 9. In worked example 1, how long does it take for the computer to depreciate to a value of $0? The computer is said to be written off when it reaches this value. The other method of depreciation used is the declining balance method of depreciation. Here, the value of the item depreciates each year by a percentage of its current value. Under such depreciation, the value of the item never actually becomes zero. This type of depreciation is an example of exponential decay that we saw in chapter 9.

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WORKED Example 2

The table below shows the value of a car that is purchased new for $40 000. Age of car (years)

Value ($)

New (0)

40 000

1

32 000

2

25 600

3

20 480

4

16 384

5

13 107

Plot the points on a set of axes and graph the depreciation of the car. Use the graph to estimate the value of the car after 10 years.

1

2 3

WRITE

Draw a set of axes with age on the horizontal axis and value on the vertical. Plot the points from the table. Join the points with a smooth curve.

Value ($)

THINK

40 000 35 000 30 000 25 000 20 000 15 000 10 000 5 000 0 0 1 2 3 4 5 6 7 8 9 10 Age (years)

4

Estimate the value after 10 years from the graph you have drawn.

From the graph, the approximate value of the car after 10 years is $4000.

remember 1. Depreciation is the loss in the value of an item over time. 2. Depreciation can be of two types: (a) Straight line depreciation. The item loses a constant amount of value each year (b) Declining value depreciation. The value of an item depreciates by a percentage of its value each year. 3. Straight line depreciation can be graphed using a linear function in which the new value of the item is the vertical intercept and the gradient is the negative of the annual loss in value. 4. Declining value depreciation is an example of exponential decay and is graphed with a smooth curve.

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SkillS

10A HEET

10.1

WORKED

Example

1

SkillS

Graphing linear equations HEET

10.2 Graphing exponential functions

Modelling depreciation

1 The table below shows the depreciating value of a tractor. Age (years)

Value ($)

New (0)

100 000

1

90 000

2

80 000

3

70 000

4

60 000

5

50 000

a Draw a graph of the value of the tractor against the age of the tractor. b Write a function for the value of the tractor. 2 The table below shows the depreciating value of a tow truck. Age (years)

Value ($)

New (0)

50 000

1

42 000

2

34 000

3

26 000

4

18 000

5

10 000

Draw a graph of value against age; hence, write a value as a linear function of age. 3 The function V = 50 000 6000A shows the value, V, of a car when it is A years old. a Draw a graph of this function. b Use the graph to calculate the value of the car after 5 years. c After how many years would the car be written off? 4 A computer is bought new for $6400 and depreciates at the rate of $2000 per year. a Write a function for the value, V, of the computer against its age, A. b Draw the graph of this function. c After how many years does the computer become written off?

Chapter 10 Depreciation

WORKED

Example

2

305

5 The table below shows the declining value of a new motorcycle. Age (years)

Value ($)

New (0)

20 000

1

15 000

2

11 250

3

8 450

4

6 350

5

4 750

a Plot the points shown by the table, and draw a graph of the value of the motorcycle against age. b Use your graph to estimate the value of the motorcycle after 8 years. Give your answer correct to the nearest $1000. 6 The table below shows the declining value of a semi-trailer. Age (years)

Value ($)

New (0)

600 000

1

420 000

2

295 000

3

205 000

4

145 000

5

100 000

a Plot the points as given in the table, and then draw a curve of best fit to graph the depreciation of the semi-trailer. b Use your graph to estimate the value of the semi-trailer after 10 years. c After what number of years will the value of the semi-trailer fall below $50 000? 7 a A gymnasium values its equipment at $200 000. Each year the value of the equipment depreciates by 20% of the value of the previous year. Calculate the value of the equipment after: i 1 year ii 2 years iii 3 years iv 4 years. b Plot these points on a set of axes and draw a graph of the value of the equipment against its age.

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8 multiple choice Which of the tables below shows a straight line depreciation? A

C

Age (years)

Value ($)

Age (years)

Value ($)

New (0)

4000

New (0)

4000

1

3600

1

3600

2

3240

2

3200

3

2916

3

2800

4

2624

4

2400

5

2362

5

2000

Age (years)

Value ($)

Age (years)

Value ($)

New (0)

4000

New (0)

4000

1

3600

1

3000

2

3300

2

2500

3

3100

3

1500

4

3000

4

1000

5

2950

5

500

B

D

9 A car is bought new for $30 000. a The straight line method of depreciation sees the car lose $4000 in value each year. Complete the table below. Age (years)

Value ($)

New (0)

30 000

1 2 3 4 5 b Draw a graph of this depreciation.

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307

c The declining balance method of depreciation sees the value of the car fall by 20% of the previous year’s value. Complete the table below. Age (years)

Value ($)

New (0)

30 000

1 2 3 4 5 d On the same set of axes draw a graph of this depreciation. e After how many years is the car worth more under declining balance depreciation than under straight line depreciation?

Straight line depreciation We have already seen that the method of straight line depreciation is where the value of an item depreciates by a constant amount each year. The depreciated value of an item is called the salvage value, S. The salvage value of an asset can be calculated using the formula: S = V0 Dn where V0 is the purchase price of the asset, D is the amount of depreciation apportioned per period and n is the number of periods.

WORKED Example 3 A laundry buys dry-cleaning equipment for $30 000. The equipment depreciates at a rate of $2500 per year. Calculate the salvage value of the equipment after 6 years.

THINK 1 2 3

WRITE

Write the formula. Substitute the values of V0, D and n. Calculate the value of S.

S = V0 Dn S = $30 000 S = $15 000

$2500

6

By solving an equation we are able to calculate when the value of an asset falls below a particular amount.

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WORKED Example 4

A plumber purchases equipment for a total of $60 000. The value of the equipment is depreciated by $7500 per year. When the value of the equipment falls below $10 000 it should be replaced. Calculate the number of years after which the equipment should be replaced. THINK

WRITE

1

Write the formula.

2

Substitute for S, V0 and D.

10 000 = 60 000

3

Solve the equation to find the value of n.

7500n = 50 000 n = 6 2---

4

Give a written answer, taking the value of n up to the next whole number.

The equipment must be replaced after 7 years.

S = V0

Dn 7500n

3

Graphics Calculator tip! Solving a depreciation equation Your graphics calculator can be used as shown below to solve equations such as that arising in worked example 4. 1. From the MENU select EQUA.

2. Press F3 (Solver).

3. Delete any existing equation, and enter the equation that arises after the substitution is made. Note: You may have a different value of N, but at this stage this can be ignored. 4. Press F6 (SOLV) to solve the equation.

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remember 1. Straight line depreciation occurs when the value of an asset depreciates by a constant amount each year. 2. The formula to calculate the salvage value, S, of an asset is: S = V0 Dn where V0 is the purchase price of the asset, D is the amount of depreciation apportioned per period and n is the total number of periods. 3. To calculate a value of V0, D or n we substitute all known values and solve the equation that is formed.

10B WORKED

Example

1 A car that is purchased for $45 000 depreciates by $5000 each year. Calculate the salvage value of the car after 5 years.

10.3 SkillS

Solving 2 Calculate the salvage value: linear a after 5 years of a computer that is purchased for $5000 and depreciates by $800 equations per year b after 7 years of a motorbike that is purchased for $25 000 and depreciates by $2100 per year c after 6 years of a semi-trailer that is purchased for $750 000 and depreciates by $80 000 per year d after 2 years of a mobile phone that is purchased for $225 and depreciates by $40 per year e after 4 years of a farmer’s plough that is purchased for $80 000 and depreciates by $12 000 per year.

3 A bus company buys 15 buses for $475 000 each. a Calculate the total cost of the fleet of buses. b If each bus depreciates by $25 000 each year, calculate the salvage value of the fleet of buses after 9 years. 4 The price of a new car is $25 000. The value of the car depreciates by $300 each month. Calculate the salvage value of the car after 4 years. WORKED

Example

4

5 An aeroplane is bought by an airline for $600 million. If the aeroplane depreciates by $40 million each year, calculate when the value of the aeroplane falls below $300 million. 6 Calculate the length of time for each of the following items to depreciate to the value given. a A computer purchased for $5600 to depreciate to less than $1000 at $900 per year b An electric guitar purchased for $1200 to depreciate to less than $500 at $150 per year c An entertainment unit purchased for $6000 to become worthless at $750 per year d Office equipment purchased for $12 000 to depreciate to less than $2500 at $1500 per year

HEET

3

Straight line depreciation

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7 A motor vehicle depreciates from $40 000 to $15 000 in 10 years. Assuming that it is depreciating in a straight line, calculate the annual amount of depreciation. 8 Calculate the annual amount of depreciation in an asset that depreciates: a from $20 000 to $4000 in 4 years b from $175 000 to $50 000 in 10 years c from $430 000 to $299 500 in 9 years. 9 A computer purchased for $3600 is written off in 4 years. Calculate the annual amount of depreciation.

Work

10 A car that is 5 years old has an insured value of $12 500. If the car is depreciating at a rate of $2500 per year, calculate its purchase price.

T SHEE

11 Calculate the purchase price of each of the following assets given that: a after 5 years the value is $50 000 and is depreciating at $12 000 per year b after 15 years the value is $4000 and is depreciating at $1500 per year c after 25 years the value is $200 and is depreciating at $50 per year.

10.1

12 An asset that depreciates at $6500 per year is written off after 12 years. Calculate the purchase price of that asset.

Declining balance method of depreciation The declining balance method of depreciation occurs when the value of an asset depreciates by a given percentage each period. Consider the case of a car purchased new for $30 000, which depreciates at the rate of 20% p.a. Each year the salvage value of the car is 80% of its value at the end of the previous year. After 1 year: S = 80% of $30 000 = $24 000 After 2 years: S = 80% of $24 000 = $19 200 After 3 years: S = 80% of $19 200 = $15 360

WORKED Example 5 A small truck that was purchased for $45 000 depreciates at a rate of 25% p.a. By calculating the value at the end of each year, find the salvage value of the truck after 4 years. THINK 1

2

The salvage value at the end of each year will be 75% of its value at the end of the previous year. Find the value after 1 year by calculating 75% of $45 000.

WRITE

After 1 year: S = 75% of $45 000 = $33 750

Chapter 10 Depreciation

THINK 3 4 5

311

WRITE

Find the value after 2 years by calculating 75% of $33 750. Find the value after 3 years by calculating 75% of $25 312.50. Find the value after 4 years by calculating 75% of $18 984.38.

After 2 years: S = 75% of $33 750 = $25 312.50 After 3 years: S = 75% of $25 312.50 = $18 984.38 After 4 years: S = 75% of $18 984.38 = $14 238.28

The salvage value under a declining balance can be calculated using the formula: S = V0(1

r)n

where S is the salvage value, V0 is the purchase price, r is the percentage depreciation per period expressed as a decimal and n is the number of periods. This formula can be considered as being similar to the compound interest formula. In the case of depreciation, however, you need to subtract rather than add the depreciation expressed as a decimal from 1.

WORKED Example 6 The purchase price of a boat is $15 000. The value of the boat depreciates by 10% p.a. Calculate the salvage value of the boat after 8 years. THINK 1 2 3

WRITE

Write the formula. Substitute values for V0, r and n. Calculate the salvage value.

S = V0(1 r)n S = $15 000 0.98 S = $6457.00

To calculate the amount by which the asset has depreciated, we subtract the salvage value from the purchase price.

WORKED Example 7 The purchase price of a motor vehicle is $40 000. The vehicle depreciates by 12% p.a. Calculate the amount by which the vehicle depreciates in 10 years. THINK 1 2 3 4

Write the formula. Substitute the value of V0, r and n. Calculate the value of S. Calculate the amount of depreciation by subtracting the salvage value from the purchase price.

WRITE S = V0(1 r)n S = $40 000 0.8810 S = $11 140.04 Depreciation = $40 000 $11 140.04 Depreciation = $28 859.96

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remember 1. The declining method of depreciation occurs when the value of an asset depreciates by a fixed percentage each year. 2. The salvage value of an asset can be calculated by subtracting the percentage depreciation each year. 3. The salvage value can be calculated using the formula: S = V0(1 r)n where S is the salvage value, V0 is the purchase price, r is the percentage depreciation per period expressed as a decimal and n is the number of periods. 4. To calculate the amount of depreciation, the salvage value should be subtracted from the purchase price.

SkillS

10C HEET

10.4

WORKED

Example

5

SkillS

Calculating compound interest HEET

Declining balance method of depreciation

1 The purchase price of a forklift is $50 000. The value of the forklift depreciates by 20% p.a. By calculating the value of the forklift at the end of each year, find the salvage value of the forklift after 4 years. 2 A trailer is purchased for $5000. The value of the trailer depreciates by 15% each year. By calculating the value of the trailer at the end of each year, calculate: a the salvage value of the trailer after 5 years (to the nearest $10) b the amount by which the trailer depreciates: i in the first year ii in the fifth year.

10.5 Finding a percentage of a quantity (money)

3 A company purchases a mainframe computer for $3 000 000. The value of the computer depreciates by 15% p.a. By calculating the value at the end of each year, find the number of years that it takes for the salvage value of the mainframe to fall below $1 000 000. WORKED

Example

6

4 Use the formula S = V0(1 r)n to calculate the salvage value after 7 years of a power generator purchased for $800 000 that depreciates at a rate of 10% p.a. (Give your answer correct to the nearest $1000.) 5 Calculate the salvage value of an asset (correct to the nearest $100) with a purchase price of: a $10 000 that depreciates at 10% p.a. for 5 years b $250 000 that depreciates at 15% p.a. for 8 years c $5000 that depreciates at 25% p.a. for 5 years d $2.2 million that depreciates at 30% p.a. for 10 years e $50 000 that depreciates at 40% p.a. for 5 years.

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313

6 A plumber has tools and equipment valued at $18 000. If the value of the equipment depreciates by 30% each year, calculate the value of the equipment after 3 years. WORKED

Example

7

7 A yacht is valued at $950 000. The value of the yacht depreciates by 22% p.a. Calculate the amount that the yacht will depreciate in value over the first 5 years (correct to the nearest $1000). 8 A new car is purchased for $35 000. The owner plans to keep the car for 5 years and then trade the car in on another new car. The estimate is that the value of the car will depreciate by 16% p.a. Calculate: a the amount the owner can expect as a trade in for the car in 5 years (correct to the nearest $100) b the amount by which the car will depreciate in 5 years. 9 multiple choice A shop owner purchases fittings for her store that cost a total of $120 000. Three years later, the shop owner is asked to value the fittings for insurance. If the shop owner allows for depreciation of 15% on the fittings, which of the following calculations will give the correct estimate of their value? A 120 000 × 0.853 B 120 000 × 0.153 C 120 000 × 0.55 D 120 000 × 0.45 10 multiple choice A computer purchased for $3000 will depreciate by 25% p.a. The salvage value of the computer after 4 years will be closest to: A $0 B $10 C $950 D $2000 11 An electrician purchases tools of trade for a total of $8000. Each year the electrician is entitled to a tax deduction for the depreciation of this equipment. If the rate of depreciation allowed is 33%, calculate: a the value of the equipment at the end of one year (correct to the nearest $1) b the tax deduction allowed in the first year c the value of the equipment at the end of two years (correct to the nearest $1) d the tax deduction allowed in the second year. 12 An accountant purchased a computer for $6000. The value of the computer depreciates by 33% p.a. When the value of the computer falls below $1000, it is written off and a new one is purchased. How many years will it take for the computer to be written off?

Rates of depreciation In the previous investigation you chose a make and model of car and researched the salvage value of this car after each year. 1 Calculate the percentage depreciation for each year. 2 Calculate if this percentage rate is approximately the same each year. 3 Using the average annual depreciation, calculate a table of salvage values for the first 5 years of the car’s life. 4 Draw a graph showing the depreciating value of the car.

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1 1 The price of a new DVD player is $1250. The player will depreciate under straight line depreciation at a rate of $200 per year. Calculate the value of the player after 3 years. 2 An asset that was valued at $39 000 when new depreciates to $22 550 in 7 years. Calculate the annual amount of depreciation under straight line depreciation. 3 A computer that is purchased new for $9000 depreciates at a rate of $1350 per year. Calculate the length of time before the computer is written off. 4 A car dealer values a used car at $7000. If the car is 8 years old and the rate of depreciation is $1750 per year, calculate the value of the car when new. 5 Write the formula for depreciation under the declining balance method. 6 A truck is valued new at $50 000 and depreciates at a rate of 32% p.a. Calculate the value of the truck after 5 years (correct to the nearest $50). 7 An asset that has a purchase price of $400 000 depreciates at a rate of 45% p.a. Calculate the asset’s value after 6 years (correct to the nearest $1000). 8 For the asset in question 7, calculate the amount by which it has depreciated in 6 years. 9 Office equipment valued at $250 000 depreciates at a rate of 15% p.a. Calculate the amount by which it depreciates in the first year. 10 Calculate the length of time it will take for the salvage value of the office equipment in question 9 to fall below $20 000.

Depreciation tables The computer application below will prepare a table that will show the depreciated value of an asset with a purchase price of $1 over various periods of time and various rates of depreciation.

Computer Application 1 Depreciation table 1. Open a new spreadsheet and enter the following information. 2. In cell B3 enter the formula =(1-B$2)^$A3. 3. Highlight the range of cells B3 to J12. Then use the Edit and then the Fill and Right and Fill and Down functions to copy the formula throughout the table. 4. The table that you now have should have the values shown in the table on page 315.

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315

Rate of depreciation (per annum) Time (years)

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1

0.9500 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000

2

0.9025 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500

3

0.8574 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250

4

0.8145 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625

5

0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0313

6

0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156

7

0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078

8

0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039

9

0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020

10

0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010 5. Use the spreadsheets’ graphing facility to draw a depreciation graph for each of the depreciation rates shown in the table. The table produced by the computer application shows the depreciated value of $1 and can be used to make calculations about depreciation.

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WORKED Example 8 An item is purchased for $500 and depreciates at a rate of 15% p.a. Use the depreciation table on page 315 to calculate the value of the item after 4 years. THINK 1

2

WRITE

Look up the table to find the depreciated value of $1 at 15% p.a. for 4 years. Multiply the depreciated value of $1 by $500.

Depreciated value = 0.5220 Depreciated value = $261

$500

The computer application on pages 314–15 will produce a general table for a declining balance depreciation. We should be able to use the formula to create a table and graph showing the salvage value of an asset under both straight line and declining balance depreciation.

WORKED Example 9 A car is purchased new for $20 000. The depreciation can be calculated under straight line depreciation at $2500 per year and under declining balance at 20% p.a. a Complete the table below. (Give all values to the nearest $1.) Age of car (years)

Straight line value ($)

Declining balance value ($)

New (0)

20 000

20 000

1 2 3 4 5 6 7 8 b Draw a graph of both the straight line and declining balance depreciation and use the graph to show the point at which the straight line value of the car falls below the declining balance value. THINK a

1 2

Copy the table. Complete the straight line column by subtracting $2500 from the previous year’s value.

WRITE

Chapter 10 Depreciation

3

b

1

2

3

WRITE

Complete the declining balance by multiplying the previous year’s value by 0.8.

Plot the points generated by the table. Join the points for the straight line depreciation with a straight line. Join the points for the declining balance depreciation with a smooth curve.

a Age of car (years)

Straight line value ($)

Declining balance value ($)

New (0)

20 000

20 000

1

17 500

16 000

2

15 000

12 800

3

12 500

10 240

4

10 000

8 192

5

7 500

6 554

6

5 000

5 243

7

2 500

4 194

8

0

3 355

b 25 000 Value ($)

THINK

317

20 000

Straight line value Declining balance value

15 000 10 000 5 000 0 (New)0 1 2 3 4 5 6 7 8 Age (years)

4

The graph shows the straight line going below the curve after 6 years.

The straight line depreciation value becomes less than the declining balance depreciation value after 6 years.

Depreciation is an allowable tax deduction for people in many occupations. A tax deduction for depreciation is allowed when equipment used in earning an income depreciates in value and will eventually need replacing. Depending on the equipment and the occupation, either straight line or declining balance depreciation may be used. Under declining balance depreciation, when the salvage value falls below a certain point the equipment may be written off. This means that the entire remaining balance can be claimed as a tax deduction and as such is considered worthless. From this point on, no further tax deductions can be claimed for this equipment.

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WORKED Example 10 A builder has tools of trade that are purchased new for $14 000. He is allowed a tax deduction of 33% p.a. for depreciation of this equipment. When the salvage value of the equipment falls below $3000, the builder is allowed to write the equipment off on the next year’s return. Complete the depreciation table below. (Use whole dollars only.) Years

Salvage value ($)

Tax deduction ($)

1 2 3 4 5 THINK 1

2

3

WRITE

Calculate the salvage value by multiplying the previous year’s value by 0.67. Calculate the tax deduction by multiplying the previous year’s value by 0.33. When the salvage value is less than $3000, claim the entire amount as a tax deduction.

Year

Salvage value ($)

Tax deduction ($)

1

9380

4620

2

6285

3095

3

4211

2074

4

2821

1390

5

0

2821

remember 1. Graphs can be drawn to compare the salvage value of an asset under different rates of depreciation, or to compare declining balance and straight line depreciation. 2. The amount by which an asset depreciates can, in many cases, be claimed as a tax deduction.

Chapter 10 Depreciation

10D

319

Depreciation tables

1 Use the table of depreciated values of $1 to calculate: a the value of a computer purchased for $5000 after 5 years, given that it depreci8 ates at 20% p.a. b the value of a car after 8 years with an initial value of $35 000, given that it depreciates at 15% p.a. c the value of a boat with an initial value of $100 000 after 10 years, given that it depreciates at 10% p.a.

WORKED

10.5 SkillS

Example

HEET

Finding a percentage of a quantity (money)

Example

SkillS

HEET

2 A taxi owner purchases a new taxi for $40 000. The taxi depreciates under straight line 10.6 depreciation at $5000 per year and under declining balance depreciation at 20% p.a. 9 Reading a Copy and complete the table below. Give all values to the nearest $100.

WORKED

financial tables

Straight line value ($)

Declining balance value ($)

New (0)

40 000

40 000

10.7 SkillS HEET

Age of car (years)

Increase or decreasee by a percentage

1 2 3 4 5 6 7 8 b Draw a graph of the salvage value of the taxi under both methods of depreciation. c State when the value under straight line depreciation becomes less than under declining balance depreciation. 3 A company has office equipment that is valued at $100 000. The value of the equipment can be depreciated at $10 000 each year or by 15% p.a. a Draw a table that will show the salvage value of the office equipment for the first ten years using both methods. (Give all values correct to the nearest $50.) b Draw a graph of the depreciating value of the equipment under both methods of depreciation.

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4 A computer purchased new for $4400 can be depreciated at either 20% p.a. or 35% p.a. Draw a table and a graph that compare the salvage value of the computer at each rate of depreciation over a 6-year period. 5 A teacher purchases a laptop computer for $6500. A tax deduction for depreciation of the computer is allowed at the rate of 33% p.a. When the value of the computer falls 10 below $1000, the computer can be written off. Copy and complete the table below. (Give all values correct to the nearest $1.)

WORKED

Example

Year

Salvage value ($)

Tax deduction ($)

1 2 3 4 5 6 6 A plumber purchases a work van for $45 000. The van can be depreciated at a rate of 25% p.a. for tax purposes, and the van can be written off at the end of 8 years. Copy and complete the depreciation schedule below. (Give all answers correct to the nearest $1.) Year 1 2 3 4 5 6 7 8

Salvage value ($)

Tax deduction ($)

Chapter 10 Depreciation

321

7 A truck is purchased for $250 000. The truck can be depreciated at the rate of $25 000 each year or over 10 years at 20% p.a. a Copy and complete the table below. (Give all values correct to the nearest $1.) Age of truck (years)

Straight line value ($)

Declining balance value ($)

New (0)

250 000

250 000

1 2 3 4 5 6 7 8 9 10 b Draw a graph of the depreciating value of the truck under both methods of depreciation. c Complete a depreciation schedule for each method of calculation.

9 Calculate the amount of depreciation on each of the following assets. a A tractor with an initial value of $80 000 that depreciates at 15% p.a. for 3 months b A bicycle with an initial value of $600 that depreciates at 25% p.a. for 6 months c Office furniture with an initial value of $8000 that depreciates at 30% p.a. for 8 months d A set of encyclopedias with an initial value of $2500 that depreciates at 40% p.a. for 9 months

Work

8 Tony is a plumber and on 1 March purchases a panel van for work purposes. The cost of the panel van is $40 000, and for tax purposes the panel van depreciates at the rate of 25% p.a. a Calculate the amount that the panel van will depreciate in the first year. b The financial year ends on 30 June. For what fraction of the financial year did Tony own the panel van? c Tony is allowed a tax deduction for depreciation of his work van. Calculate the amount of tax deduction that Tony is allowed for the financial year ending on 30 June.

T SHEE

10.2

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summary Modelling depreciation • Depreciation can be calculated in two ways. The depreciation can be straight line depreciation or declining balance depreciation. • Straight line depreciation occurs when the value of an asset decreases by a constant amount each year. The graph of the salvage value is a straight line, the vertical intercept is the purchase price and the gradient is the negative of the annual depreciation. • Declining balance depreciation occurs when the salvage value of the item is a percentage of the previous year’s value. The graph of a declining balance depreciation will be an exponential decay graph.

Straight line depreciation • The salvage value of an asset under straight line depreciation can be calculated using the formula: S = V0 Dn where S is the salvage value, V0 is the purchase price of the asset, D is the amount of depreciation apportioned per period and n is the number of periods of depreciation. • Values of V0, D or n can be calculated by substitution and solving the equation formed.

Declining balance depreciation • Under declining balance depreciation the salvage value of an asset can be calculated using the formula: S = V0(1 r)n where r is the percentage depreciation per period expressed as a decimal. • To calculate the amount by which an asset depreciates in a year, we subtract the salvage value at the end of the year from the salvage value at the beginning of the year.

Depreciation tables • Depreciation can be compared using either a table or a graph. • Tax deductions are allowed for depreciation of assets that are used as part of earning an income. • A depreciation schedule is used to calculate tax deductions over a period of years on an asset.

Chapter 10 Depreciation

323

CHAPTER review 1 The table below shows the depreciating value of a pleasure cruiser. Age (years)

Value ($)

New (0)

200 000

1

180 000

2

160 000

3

140 000

4

120 000

5

100 000

10A

a Draw a graph of the value of the pleasure cruiser against its age. b Write a function for the value of the pleasure cruiser. 2 The table below shows the depreciating value of a racing bike. Age (years)

Value ($)

New (0)

3500

1

3250

2

3000

3

2750

4

2500

5

2250

10A

a Draw a graph of the value of the bike against age. b Write a function for the straight line depreciation. c Use your graph to estimate the value of the bike after 9 years. 3 The function V = 15 000 900A shows the value, V, of a motorcycle when it is A years old. a Draw a graph of this function. b Use the graph to calculate the value of the motorcycle after 5 years. c After how many years would the motorcycle be written off?

10A

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Maths Quest General Mathematics HSC Course

4 The table below shows the declining value of a delivery van. Age (years)

Value ($)

New (0)

60 000

1

48 000

2

38 400

3

30 720

4

24 576

5

19 660

a Plot the points as given in the table, and then draw a curve of best fit to graph the depreciation of the van. b Use your graph to estimate the value of the van after 10 years. c After what number of years will the value of the van fall below $10 000?

10A

5 A laundry buys dry-cleaning equipment for $8000. Each year the equipment depreciates by 25% of the previous year’s value. Calculate the value of the equipment at the end of the first five years, and use the results to draw a graph of the depreciation.

10B

6 The purchase price of a car is $32 500. The car depreciates by $3250 each year. Use the formula S = V0 Dn to calculate the salvage value of the car after 8 years.

10B

7 Calculate the salvage value of an asset: a after 6 years, that was purchased for $4000 and depreciates by $450 each year b after 10 years, that was purchased for $75 000 and depreciates by $6000 each year c after 9 years, that was purchased for $640 000 and depreciates by $45 000 each year.

10B

8 A movie projector is purchased by a cinema for $30 000. The projector depreciates by $2500 each year. Calculate the length of time it takes for the projector to be written off.

10B

9 A camera that was purchased new for $1500 has a salvage value of $500 four years later. Calculate the annual amount of depreciation on the camera.

10B

10 Arthur buys a car for $25 000. The depreciation on the car is $2250 each year. He decides that he will trade the car in on a new car in the final year before the salvage value falls below $10 000. When will Arthur trade the car in?

10C

11 The purchase price of a mobile home is $40 000. The value of the mobile home depreciates by 15% p.a. By calculating the value of the mobile home at the end of each year, find the salvage value of the mobile home after 4 years. (Give your answer correct to the nearest $1.)

10C

12 Use the formula S = V0(1 r)n to calculate the salvage value after 7 years of a crop duster that was purchased for $850 000 and depreciates at 8% p.a. (Give your answer correct to the nearest $1000.)

10C

13 Calculate the salvage value of an asset (correct to the nearest $10) with a purchase price of: a $40 000 that depreciates at 10% p.a. for 5 years b $1500 that depreciates at 4% p.a. for 10 years c $180 000 that depreciates at 12.5% p.a. for 15 years d $4.5 million that depreciates at 40% p.a. for 10 years e $250 000 that depreciates at 33 1--- % p.a. for 4 years. 3

Chapter 10 Depreciation

325

14 A company buys a new bus for $600 000. The company keeps buses for 10 years and then trades them in on a new bus. The estimate is that the value of the bus will depreciate by 12% p.a. Calculate: a the amount the owner can expect as a trade-in for the bus in 10 years b the amount by which the bus will depreciate in 10 years.

10C

15 A company has office equipment that is valued at $100 000. The value of the equipment can be depreciated at $10 000 each year or by 15% p.a. a Draw a table to show the salvage value of the office equipment for the first ten years. b Draw a graph of the depreciating value of the equipment under both depreciation methods.

10D

16 A personal computer is purchased for $4500. A tax deduction for depreciation of the computer is allowed at the rate of 33% p.a. When the value of the computer falls below $1000, the computer can be written off. Copy and complete the table below.

10D

Year

Salvage value ($)

Tax deduction ($)

1 2 3 4 5

Practice examination questions 1 multiple choice Which of the following tables gives an example of declining balance depreciation? A

C

Year

Salvage value

Year

Salvage value ($)

New (0)

20 000

New (0)

20 000

1

18 000

1

18 200

2

16 200

2

16 400

3

14 580

3

14 600

4

13 122

4

12 800

Year

Salvage value

Year

Salvage value ($)

New (0)

20 000

New (0)

20 000

1

18 000

1

17 000

2

16 500

2

15 000

3

15 500

3

14 000

4

15 000

4

13 500

B

D

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2 multiple choice A helicopter is purchased by a company for $3.3 million. The salvage value of the helicopter depreciates in a straight line at a rate of $240 000 per year. After how many years will the value of the helicopter be less than $1 million? A 8 B 9 C 10 D 11 3 multiple choice Trevor purchases a new computer for $5000. It depreciates under declining balance depreciation at a rate of 20% p.a. Each year Trevor claims the amount of depreciation on the computer as a tax deduction. The amount of Trevor’s tax deduction in the third year is: A $640 B $1000 C $2560 D $3200 4 multiple choice The value of a new car depreciates by 12.5% p.a. The salvage value in 5 years of a car that was purchased new for $37 500 is (to the nearest $100): A $9375 B $18 300 C $19 200 D $32 800 5 The value of a home theatre system when purchased new is $3000. The system depreciates at the rate of 15% p.a. under declining balance depreciation. a Calculate the salvage value of the system in 4 years (correct to the nearest $1). b By how much has the system depreciated in this time? c Calculate the equivalent rate of straight line depreciation over the four years. d Graph the salvage value of the home theatre system under both declining balance and straight line depreciation. 6 An office is fitted with $200 000 of office equipment. The company claims tax deductions for the depreciation of the equipment at the rate of 12% p.a. a Calculate the amount of tax deduction claimed by the company in the first year. b Complete the depreciation schedule below. Year

Salvage value ($)

Tax deduction ($)

1 2 3 4

CHAPTER

test yourself

10

5 c When the value of the equipment falls below $50 000, the equipment is written off and replaced. After how many years will the equipment be written off?

The normal distribution

11 syllabus reference Data analysis 6 • The normal distribution

In this chapter 11A z-scores 11B Comparison of scores 11C Distribution of scores

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

11.1

11.2

Finding the mean

1 Find the mean of each of the following data sets. a 4, 6, 2, 7, 9, 3, 6, 9 b 45, 72, 90, 70, 69, 48, 62, 99, 47, 55 c d Score Frequency Score Frequency 6

3

15

12

7

13

16

36

8

16

17

42

9

12

18

15

10

6

19

11

20

9

Finding the standard deviation

2 Find the population standard deviation of each of the data sets in question 1. Give each answer correct to 1 decimal place.

11.3

Choosing the appropriate standard deviation

3 In each of the following choose the appropriate measure of standard deviation. a On entering a certain music store people are asked how many CDs they own. b The number of parliamentarians who vote in favour of bills brought before parliament.

Chapter 11 The normal distribution

329

z-scores A normal distribution is a statistical occurrence where a data set of scores is symmetrically distributed about the mean. Most continuous variables in a population, such as height, mass and time, are normally distributed. In a normal distribution, the frequency histogram is symmetrical and begins to take on a x– bell shape as shown by the figure on the right. The normal distribution is symmetrical about the mean, which has the same value as the median and mode in this distribution. The graph of a normal distribution will extend symmetrically in both directions and will always remain above the x-axis. The spread of the normal distribution will depend on the standard deviation. The lower the standard deviation, the more clustered the scores will be around the mean. The figure below left shows a normal distribution with a low standard deviation, while the figure below right shows a normal distribution with a much greater standard deviation.

x– x–

To gain a comparison between a particular score and the rest of the population, we use the z-score. The z-score (or standardised score) indicates the position of a particular score in relation to the mean. z-scores are a very important statistical measure and later in the chapter some of their uses will be explained. A z-score of 0 indicates that the score obtained is equal to the mean, a negative z-score indicates that the score is below the mean and a positive z-score indicates a score above the mean. The z-score measures the distance from the mean in terms of the standard deviation. A score that is exactly one standard deviation above the mean has a z-score of 1. A score that is exactly one standard deviation below the mean has a z-score of 1. To calculate a z-score we use the formula: x–x z = ----------s where x is the score, x is the mean and s is the standard deviation.

WORKED Example 1

In an IQ test the mean IQ is 100 and the standard deviation is 15. Dale’s test results give an IQ of 130. Calculate this as a z-score. THINK 1

Write the formula.

2

Substitute for x, x and s.

3

Calculate the z-score.

WRITE x–x z = ----------s 130 – 100 z = -----------------------15 z=2

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Maths Quest General Mathematics HSC Course

Dale’s z-score is 2, meaning that his IQ is exactly two standard deviations above the mean. Not all z-scores will be whole numbers; in fact most will not be whole numbers. A whole number indicates only that the score is an exact number of standard deviations above or below the mean.

WORKED Example 2 A sample of professional basketball players gives the mean height as 192 cm with a standard deviation of 12 cm. Dieter is 183 cm tall. Calculate Dieter’s height as a z-score. THINK

WRITE

1

Write the formula.

2

Substitute for x, x and s.

3

Calculate the z-score.

x–x z = ----------s 183 – 192 z = -----------------------12 z = 0.75

The negative z-score in worked example 2 indicates that Dieter’s height is below the mean but, in this case, by less than one standard deviation. When examining z-scores, care must be taken to use the appropriate value for the standard deviation. If examining a population, the population standard deviation ( n) should be used and if a sample has been taken, the sample standard deviation ( n 1 or sn) should be used. Remember: Your graphics calculator displays all of this information once data is stored and calculated using the statistics function.

WORKED Example 3 To obtain the average number of hours study done by students in her class per week, Kate surveys 20 students and obtains the following results. 12 18 15 14 9 10 13 12 18 25 15 10 3 21 11 12 14 16 17 20 a Calculate the mean and population standard deviation (correct to 3 decimal places). b Robert does 16 hours of study each week. Express this as a z-score based on the above results. (Give your answer correct to 3 decimal places.) THINK

WRITE

a

a

1 2 3

b

Enter the data into your calculator. Obtain the mean from your calculator. Obtain the standard deviation from your calculator using the sample standard deviation.

1

Write the formula.

2

Substitute for x, x and s.

3

Calculate the z-score.

x = 14.25 sn = 4.753

x–x b z = ----------s 16 – 14.25 z = ------------------------4.753 z = 0.368

Chapter 11 The normal distribution

331

Graphics Calculator tip! Finding the z-score Your graphics calculator can be used to find a z-score once the data is stored and the calculator has in its memory the mean and standard deviation. The z-score calculated here is found using the population standard deviation. Make sure that this is the appropriate standard deviation for the question that you are doing. Consider worked example 3. 1. From the MENU select STAT.

2. Delete any existing data and enter the scores from worked example 3 in List 1.

3. Press F2 (CALC). You may need to press F6 first for more options.

4. Press F6 (SET). Check that 1Var Xlist is set to List 1 and 1Var Freq is set to 1.

5. Press EXE to return to the previous screen, and then press F1 (1VAR) to display the summary statistics.

6. Press MENU and then select RUN.

7. Press OPTN (PROB).

F6 for more options and then F3

8. Again press F6 for more options and then F4 t(). This is the z-score function, so enter 16, close brackets and press EXE .

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Maths Quest General Mathematics HSC Course

remember 1. A data set is normally distributed if it is symmetrical about the mean. 2. The graph of a normally distributed data set is a bell-shaped curve that is symmetrical about the mean. In such a distribution the mean, median and mode are equal. 3. A z-score is used to measure the position of a score in a data set relative to the mean. 4. The formula used to calculate a z-score is: x–x z = ----------s where x is the score, x is the mean, and s is the standard deviation.

SkillS

11A HEET

11.1

WORKED

Example

1

SkillS

3 Tracy is a nurse and samples the mass of 50 newborn babies born in the hospital in which she works. She finds that the mean mass is 3.5 kg, with a standard deviation of 0.4 kg. What would be the standardised score of a baby whose birth mass was: a 3.5 kg? b 3.9 kg? c 2.7 kg? d 4.7 kg? e 3.1 kg?

11.2

SkillS

Finding the standard deviation HEET

1 In a Maths exam the mean score is 60 and the standard deviation is 12. Chifune’s mark is 96. Calculate her mark as a z-score. 2 In an English test the mean score was 55 with a standard deviation of 5. Adrian scored 45 on the English test. Calculate Adrian’s mark on the test as a z-score.

Finding the mean HEET

z-scores

11.3 Choosing the appropriate standard deviation

4 Ricky finds that the mean number of hours spent watching television each week by Year 12 students is 10.5 hours, with a standard deviation of 3.2 hours. How many hours of television is watched by a person who has a standardised score of: a 0? b 1? c 2? d 1? e 3? WORKED

Example

2

5 IQ tests have a mean of 100 and a standard deviation of 15. Calculate the z-score for a person with an IQ of 96. (Give your answer correct to 2 decimal places.) 6 The mean time taken for a racehorse to run 1 km is 57.69 s, with a standard deviation of 0.36 s. Calculate the z-score of a racehorse that runs 1 km in 58.23 s.

Chapter 11 The normal distribution

333

7 In a major exam every subject has a mean score of 60 and a standard deviation of 12.5. Clarissa obtains the following marks on her exams. Express each as a z-score. a English 54 b Maths 78 c Biology 61 d Geography 32 e Art 95 8 The mean time for athletes over 100 m is 10.3 s, with a standard deviation of 0.14 s. What time would correspond to a z-score of: a 0? b 2? c 0.5? d −3? e −0.35? f 1.6? WORKED

Example

3

9 The length of bolts being produced by a machine needs to be measured. To do this, a sample of 20 bolts are taken and measured. The results (in mm) are given below. 20 19 18 21 20 17 19 21 22 21 17 17 21 20 17 19 18 22 22 20 a Calculate the mean and standard deviation of the distribution. b A bolt produced by the machine is 22.5 mm long. Express this result as a z-score. (Give your answer correct to 2 decimal places.) 10 A garage has 50 customers who have credit accounts with them. The amount spent by each credit account customer each week is shown in the table below. Amount ($)

Class centre

Frequency

0–20

2

20–40

8

40–60

19

60–80

15

80–100

6

a Copy and complete the table. b Calculate the mean and standard deviation. c Calculate the z-score that corresponds to a customer’s weekly account of: i $50 ii $100 iii $15.40. 11 multiple choice In a normal distribution, the mean is 21.7 and the standard deviation is 1.9. A score of 20.75 corresponds to a z-score of: A −1 B −0.5 C 0.5 D1 12 multiple choice In a normal distribution, the mean is 58. A score of 70 corresponds to a standardised score of 1.5. The standard deviation of the distribution is: A6 B 8 C 10 D 12 13 multiple choice In a normal distribution, a score of 4.6 corresponds to a z-score of –2.4. It is known that the standard deviation of the distribution is 0.8. The mean of the distribution is: A 2.2 B 2.68 C 6.52 D 6.8

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Maths Quest General Mathematics HSC Course

14 The results of 24 students sitting a Maths exam are listed below. 95 63 45 48 78 75 80 66 60 58 59 62 52 57 64 75 81 60 65 70 65 63 62 49 a Calculate the mean and standard deviation of the exam marks. b Calculate the standardised score of the highest score and the lowest score, correct to 2 decimal places.

Work

15 The results of Luke’s exams are shown in the table below.

T SHEE

11.1

Subject

Luke’s mark

Mean

Standard deviation

English

72

60

12

Maths

72

55

13

Biology

76

64

8

Computing studies

60

70

5

Visual arts

60

50

15

Music

50

58

10

Convert each of Luke’s results to a standardised score.

Comparison of scores An important use of z-scores is to compare scores from different data sets. Suppose that in your Maths exam your result was 74 and in English your result was 63. In which subject did you achieve the better result? It may appear, at first glance, that the Maths result is better, but this does not take into account the difficulty of the test. A mark of 63 on a difficult English test may in fact be a better result than 74 if it was an easy Maths test. The only way that we can fairly compare the results is by comparing each result with its mean and standard deviation. This is done by converting each result to a z-score. If for Maths x = 60 and

n

= 12, then

x–x z = ----------s 74 – 60 = -----------------12 = 1.17

And if for English x = 50 and

n

x–x = 8, then z = ----------s 63 – 50 = -----------------8 = 1.625

The English result is better because the higher z-score shows that the 63 is higher in comparison to the mean of each subject.

Chapter 11 The normal distribution

335

WORKED Example 4 Janine scored 82 in her Physics exam and 78 in her Chemistry exam. In Physics, x = 62 and n = 10, while in Chemistry, x = 66 and n = 5. a Write both results as a standardised score. b Which is the better result? Explain your answer. THINK

WRITE

a

x–x x–x a Physics: z = ----------- Chemistry: z = ----------s s 82 – 62 78 – 66 = -----------------= -----------------10 5 =2 = 2.4

1

Write the formula for each subject.

2

Substitute for x, x and s.

3

Calculate each z-score.

b Explain that the subject with the highest z-score is the better result.

b The Chemistry result is better because of the higher z-score.

In each example the circumstances must be read carefully to see whether a higher or lower z-score is better. For example, if we were comparing times for runners over different distances, the lower z-score would be the better one.

WORKED Example 5 In international swimming the mean time for the men’s 100 m freestyle is 50.46 s with a standard deviation of 0.6 s. For the 200 m freestyle, the mean time is 1 min 51.4 s with a standard deviation of 1.4 s. Sam’s best time is 49.92 s for 100 m and 1 min 49.3 for 200 m. At a competition Sam can enter only one of these events. Which event should he enter? THINK

WRITE

1

Write the formula for both events.

2

Substitute for x, x and s. (For 200 m convert time to seconds.) Calculate the z-scores. The best event is the one with the lower z-score.

3 4

x–x 100 m: z = ----------200 m: z = s 49.92 – 50.46 = --------------------------------= 0.6

x–x ----------s 109.3 – 111.4 --------------------------------1.4

= 0.9 = 1.5 The z-score for 200 m is lower, indicating that Sam’s time is further below the mean and that this is the event that he should enter.

remember 1. Scores can be compared by their z-scores as they compare the score with the mean and the standard deviation. 2. Read each question carefully to see if a higher or lower z-score is a better outcome.

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Maths Quest General Mathematics HSC Course

11B EXCE

et

reads L Sp he

One variable statistics

WORKED

Example

4

Comparison of scores

1 Ken’s English mark was 75 and his Maths mark was 72. In English the mean was 65 with a standard deviation of 8, while in Maths the mean mark was 56 with a standard deviation of 12. a Convert the mark in each subject to a z-score. b In which subject did Ken perform better? Explain your answer. 2 In the first Maths test of the year the mean mark was 60 and the standard deviation was 12. In the second test the mean was 55 and the standard deviation was 15. Barbara scored 54 in the first test and 50 in the second test. In which test did Barbara do better? Explain your answer. 3 multiple choice The table below shows the mean and standard deviation in four subjects. Subject

Mean

Standard deviation

English

60

12

Maths

65

8

Biology

62

16

Geography

52

7.5

Kelly’s marks were English 66, Maths 70, Biology 50 and Geography 55. In which subject did Kelly achieve her best result? A English B Maths C Biology D Geography 4 multiple choice The table below shows the mean and standard deviation of house prices in four Australian cities. The table also shows the cost of building the same three-bedroom house in each of the cities. City

Mean

Standard deviation

Cost

Sydney

$230 000

$30 000

$215 000

Melbourne

$215 000

$28 000

$201 000

Adelaide

$185 000

$25 000

$160 000

Brisbane

$190 000

$20 000

$165 000

In which city is the standardised cost of building the house least? A Sydney B Melbourne C Adelaide D Brisbane

Chapter 11 The normal distribution

WORKED

Example

5

337

5 Karrie is a golfer who scored 70 on course A, which has a mean of 72 and a standard deviation of 2.5. On course B, Karrie scores 69. The mean score on course B is 72 and the standard deviation is 4. On which course did Karrie play the better round? (In golf the lower score is better.) 6 Steve is a marathon runner. On the Olympic course in Sydney the mean time is 2 hours and 15 minutes with a standard deviation of 4.5 minutes. On Athens’ Olympic course the mean time is 2 hours and 16 minutes with a standard deviation of 3 minutes. In Sydney Steve’s time was 2 hours 17 minutes and in Athens his time was 2 hours 19 minutes. a Write both times as a z-score. b Which was the better performance? Explain your answer. 7 multiple choice The table below shows the mean and standard deviation of times in the 100 m by the same group of athletes on four different days. It also shows Matt’s time on each of these days. Day

Mean

Standard deviation

Matt’s time

8 Jan.

10.21

0.15

10.12

15 Jan.

10.48

0.28

10.30

22 Jan.

10.14

0.09

10.05

29 Jan.

10.22

0.12

10.11

On what day did Matt give his best performance? A 8 Jan. B 15 Jan. C 22 Jan.

D 29 Jan.

8 multiple choice In which of the following subjects did Alyssa achieve her best standardised result? Subject

Alyssa’s mark

Mean

Standard deviation

English

54

60

12

Maths

50

55

15

Biology

60

65

8

Music

53

62

9

A English

B Maths

C Biology

D Music

9 Shun Mei received a mark of 64 on her Maths exam and 63 on her Chemistry exam. To determine how well she actually did on the exams, Shun Mei sampled 10 people who sat for the same exams and the results are shown below. Maths: 56 45 82 90 41 32 65 60 55 69 Chemistry: 55 63 39 92 84 46 47 50 58 62

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Maths Quest General Mathematics HSC Course

a Calculate the mean and standard deviation for Shun Mei’s sample in each subject. b By converting each of Shun Mei’s marks to z-scores, state the subject in which she performed best. 10 Ricardo scored 85 on an entrance test for a job. The test has a mean score of 78 and a standard deviation of 8. Kory sits a similar exam and scores 27. In this exam the mean is 18 and the standard deviation is 6. Who is better suited for the job? Explain your answer.

1 1 In a normal distribution the mean is 32 and the standard deviation 6. Convert a score of 44 to a z-score. 2 In a normal distribution the mean is 1.2 and the standard deviation is 0.3. Convert a score of 0.6 to a z-score. 3 The mean of a distribution is 254 and the standard deviation is 39. Write a score of 214 as a standardised score, correct to 2 decimal places. 4 The mean mark on an exam is 62 and the standard deviation is 9.5. Convert a mark of 90 to a z-score. (Give your answer correct to 2 decimal places.) 5 Explain what is meant by a z-score of 1. 6 Explain what is meant by a z-score of –2. 7 In a distribution, the mean is 50 and the standard deviation is 10. What score corresponds to a z-score of 0? 8 In a distribution the mean score is 60. If a mark of 76 corresponds to a standardised score of 2, what is the standard deviation? 9 Cynthia scored a mark of 65 in English where the mean was 55 and the standard deviation is 8. In Maths Cynthia scored 66 where the mean was 52 and the standard deviation 10. Convert the mark in each subject to a z-score. 10 In which subject did Cynthia achieve her best result?

Comparison of subjects 1 List all the subjects that you study. Arrange the subjects in the order that you feel is from your strongest subject to your weakest. 2 List your most recent exam results in each subject. 3 From your teachers, find out the mean and standard deviation of the results in each subject. 4 Convert each of your marks to a standardised score. 5 List your subjects from best to worst based on the standardised score and see how this list compares with the initial list that you wrote.

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Distribution of scores In any normal distribution, the percentage of scores that lie within a certain number of standard deviations of the mean is always the same, provided that the sample is large enough. This is true irrespective of the values of the mean and standard deviation. In any normal distribution, approximately 68% of the values will lie within one standard deviation of the mean. This means 68% of scores will have a z-score between 1 and 1. 68% This can be shown on a normal curve as: z –3 –2 –1

0

Approximately 95% of the values lie within 2 standard deviations, or have a z-score of between 2 and 2.

1

2

3

95% z –3 –2 –1

0

1

2

3

2

3

Approximately 99.7% of scores lie within 3 standard deviations, or have a z-score that lies between 3 and 3. 99.7% z –3 –2 –1

0

1

If we know that a random variable is approximately normally distributed, and we know its mean and standard deviation, then we can use this rule to quickly make some important statements about the way in which the data values are distributed.

WORKED Example 6 Experience has shown that the scores obtained on a commonly used IQ test can be assumed to be normally distributed with a mean of 100 and a standard deviation of 15. Approximately what percentage of the distribution lies: a between 85 and 115? b between 70 and 130? c between 55 and 145? THINK

WRITE

a

85 – 100 115 – 100 a z = --------------------z = -----------------------15 15 = –1 =1 68% of the scores will lie between 85 and 115.

1

Calculate the z-scores for 85 and 115.

2

68% of scores have a z-score between 1 and 1.

Continued over page

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Maths Quest General Mathematics HSC Course

THINK

WRITE

b

70 – 100 130 – 100 b z = --------------------z = -----------------------15 15 = –2 =2 95% of the scores will lie between 70 and 130.

c

1

Calculate the z-scores for 70 and 130.

2

95% of scores have a z-score between 2 and 2.

1

Calculate the z-scores for 55 and 145.

2

99.7% of scores have a z-score between 3 and 3.

55 – 100 145 – 100 c z = --------------------z = -----------------------15 15 = –3 =3 99.7% will lie between 55 and 145.

We can also make statements about the percentage of scores that lie in the tails of the distribution by using the symmetry of the distribution and remembering that 50% of scores will have a z-score greater than 0 and 50% will have a z-score less than 0.

WORKED Example 7

In an exam x = 60 and above 84?

n

= 12. What percentage of candidates in the exam scored

THINK

WRITE

1

Calculate 84 as a z-score.

2

Draw a sketch showing 95% of z-scores lie between 2 and 2. 5% of z-scores therefore lie outside this range. Half of these scores lie below 2 and half are above 2.

3

x–x z = ----------s 84 – 60 z = -----------------12 z=2

2.5%

95% 95% 60

4

Give a written answer.

2.5% 84

2.5% of scores are greater than 84.

Some important terminology is used in connection with this rule. We can say that if 95% of scores have a z-score between 2 and 2, then if one member of the population is chosen, that member will very probably have a z-score between 2 and 2. If 99.7% of the population has a z-score between 3 and 3, then if one member of that population is chosen, that member will almost certainly have a z-score between 3 and 3.

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WORKED Example 8 A machine produces tyres that have a mean thickness of 12 mm, with a standard deviation of 1 mm. If one tyre that has been produced is chosen at random, within what limits will the thickness of the tyre: a very probably lie? b almost certainly lie? THINK

WRITE

a

Tyre thickness will very probably have a z-score between −2 and 2. A z-score of −2 corresponds to a tyre of 10 mm thickness. A z-score of 2 corresponds to a tyre of 14 mm thickness.

a If z = – 2 If z = 2 x = x – 2s x = x + 2s = 12 – 2 × 1 = 12 + 2 × 1 = 10 = 14 A tyre chosen will very probably have a thickness of between 10 and 14 mm.

Tyre thickness will almost certainly have a z-score between −3 and 3. A z-score of −3 corresponds to a tyre of 9 mm thickness. A z-score of 3 corresponds to a tyre of 15 mm thickness.

b If z = – 3 If z = 3 x = x – 3s x = x + 35 = 12 – 3 × 1 = 12 + 3 × 1 = 9 = 15 A tyre chosen will almost certainly have a thickness of between 9 and 15 mm.

1

2

3

b

1

2

3

Because it is almost certain that a member of the data set will lie within three standard deviations of the mean, if a possible member of the data set is found to be outside this range one should suspect a problem. For example, if a machine is set to deposit 200 mL of liquid into a bottle, with a standard deviation of 5 mL, and then a bottle is found to have contents of 220 mL, one would expect there to be a problem with the settings on the machine. This knowledge of z-scores is then used in industry by the quality control department. In the above example a sample of bottles would be tested and the z-scores recorded. The percentage of z-scores between −1 and 1, −2 and 2, and −3 and 3 are checked against the above rule. If these percentages are not correct, the machinery needs to be checked for faults.

remember 1. In a normal distribution: • 68% of scores will have a z-score between −1 and 1 • 95% of scores will have a z-score between −2 and 2 • 99.7% of scores will have a z-score between −3 and 3. 2. The symmetry of the normal distribution allows us to make calculations about the percentage of scores lying within certain limits. 3. If a member of a normally distributed population is chosen, it will: • very probably have a z-score between −2 and 2 • almost certainly have a z-score between −3 and 3. 4. Any score further than three standard deviations from the mean indicates that there may be a problem with the data set.

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11C WORKED

Example

6

Distribution of scores

1 The temperature on a January day in a city is normally distributed with a mean of 26° and a standard deviation of 3°. What percentage of January days lie between: a 23° and 29°? b 20° and 32°? c 17° and 35°? 2 The marks of students sitting for a major exam are normally distributed with x = 57 and sn = 13. What percentage of marks on the exam were between: a 44 and 70? b 31 and 83? c 18 and 96? 3 The mean thickness of bolts produced by a machine is 2.3 mm, with a standard deviation of 0.04 mm. What percentage of bolts will have a thickness between 2.22 mm and 2.38 mm?

WORKED

Example

7

4 Experience has shown that the scores obtained on a commonly used IQ test can be assumed to be normally distributed with a mean µ = 100 and a standard deviation s = 15. What percentage of scores lie above 115? 5 The heights of young women are normally distributed with a mean x = 160 cm and a standard deviation sn = 8 cm. What percentage of the women would you expect to have heights: a between 152 and 168 cm? b greater than 168 cm? c less than 136 cm? 6 The age at which women give birth to their first child is normally distributed with x = 27.5 years and sn = 3.2 years. From these data we can conclude that about 95% of women have their first child between what ages? 7 Fill in the blanks in the following statements. For any normal distribution: a 68% of the values have a z-score between ___ and ___ b ___% of the values have a z-score between –2 and 2 c ___% of the values have a z-score between ___ and ___. 8 multiple choice Medical tests indicate that the amount of an antibiotic needed to destroy a bacterial infection in a patient is normally distributed with x = 120 mg and sn = 15 mg. The percentage of patients who would require more than 150 mg to clear the infection is: A 0.15% B 2.5% C 5% D 95% 9 multiple choice The mean mark on a test is 55, with a standard deviation of 10. The percentage of students who achieved a mark between 65 and 75 is: A 13.5% B 22.5% C 34% D 95% 10 In a factory, soft drink is poured into cans such that the mean amount of soft drink is 500 mL with a standard deviation of 2 mL. Cans with less than 494 mL of soft drink are rejected and not sold to the public. What percentage of cans are rejected?

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11 The distribution of IQ scores for the inmates of a certain prison is approximately normal with a mean of 85 and a standard deviation of 15. a What percentage of this prison population have an IQ of 100 or higher? b If someone with an IQ of 70 or less can be classified as mentally disabled, what percentage of the prison population could be classified as mentally disabled? 12 The distribution of blood pressures (systolic) among women of similar ages is normal with a mean of 120 (mm of mercury) and a standard deviation of 10 (mm of mercury). Determine the percentage of women with a systolic blood pressure: a between 100 and 140 b greater than 130 c between 120 and 130 d between 90 and 110 e between 110 and 150. 13 The mass of packets of chips is normally distributed with x = 100 g and n = 2.5 g. If I purchase a packet of these chips, between what limits will the mass of the packet: 8 a very probably lie? b almost certainly lie?

WORKED

Example

14 The heights of army recruits are normally distributed about a mean of 172 cm and a standard deviation of 4.5 cm. A volunteer is chosen from the recruits. The height of the volunteer will very probably lie between what limits?

16 The average mass of babies is normally distributed with a mean of 3.8 kg and a standard deviation of 0.4 kg. A newborn baby will almost certainly have a mass between what limits?

Examining a normal distribution Complete a sample of the heights or masses of 50 people. 1 Calculate the mean and the standard deviation of your sample. 2 Calculate the percentage of people whose height or mass has a standardised score of between 1 and 1. 3 Calculate the percentage of people whose height or mass has a standardised score of between 2 and 2. 4 Calculate the percentage of people whose height or mass has a standardised score of between 3 and 3.

Work

15 A machine is set to deposit a mean of 500 g of washing powder into boxes with a standard deviation of 10 g. When a box is checked, it is found to have a mass of 550 g. What conclusion can be drawn from this?

T SHEE

11.2

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summary z-scores • A data set is normally distributed if it is symmetrical about the mean. • A z-score measures the position of a score relative to the mean and standard deviation. • A z-score is found using the formula x– x–x z = ----------s where x is the score, x is the mean, and s is the standard deviation.

Comparison of scores • Standardising both scores best compares scores from different data sets. • When comparing exam marks, the highest z-score is the best result.

Distribution of scores • A data set that is normally distributed will be symmetrical about the mean. • 68% of scores will have a z-score of between 1 and 1. • 95% of scores will have a z-score between 2 and 2. A score chosen from this data set will very probably lie in this range. • 99.7% of scores will have a z-score of between 3 and 3. A score chosen from the data set will almost certainly lie within this range.

Chapter 11 The normal distribution

345

CHAPTER review 1 Measurements of the amount of acid in a certain chemical are made. The results are normally distributed such that the mean is 6.25% and the standard deviation is 0.25%. Harlan gets a reading of 5.75%. What is Harlan’s reading as a z-score?

11A

2 A set of scores is normally distributed such that x = 15.3 and n = 5.2. Convert each of the following members of the distribution to z-scores. a 15.3 b 20.5 c 4.9 d 30.9 e 10.1

11A

3 On an exam the results are normally distributed with a mean of 58 and a standard deviation of 7.5. Jennifer scored a mark of 72 on the exam. Convert Jennifer’s mark to a z-score, giving your answer correct to 2 decimal places.

11A

4 A set of scores is normally distributed with a mean of 2.8 and a standard deviation of 0.6. Convert each of the following members of the data set to z-scores, correct to 2 decimal places. a 2.9 b 3.9 c 1 d 1.75 e 1.6

11A

5 Anji conducts a survey on the water temperature at her local beach each day for a month. The results (in °C) are shown below. 20 21 19 22 21 18 17 23 17 16 22 20 20 20 21 20 21 18 22 17 16 20 20 22 19 21 22 23 24 20 a Find the mean and standard deviation of the scores. b Find the highest and lowest temperatures in the data set and express each as a z-score.

11A

6 The table below shows the length of time for which a sample of 100 light bulbs will burn.

11A

Length of time (hours)

Class centre

Frequency

0–500

3

500–1000

28

1000–1500

59

1500–2000

10

a Find the mean and standard deviation for the data set. b A further sample of five light bulbs are chosen. The length of time for which each light bulb burned is given below. Convert each of the following to a standardised score. i 1000 hours ii 1814 hours iii 256 hours iv 751 hours v 2156 hours 7 Betty sat exams in both Physics and Chemistry. In Physics the exam results showed a mean of 48 and a standard deviation of 12, while in Chemistry the mean was 62 with a standard deviation of 9. a Betty scored 66 in Physics. Convert this result to a z-score. b Betty scored 71 in Chemistry. Convert this result to a z-score. c In which subject did Betty achieve the better result? Explain your answer.

11B

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Maths Quest General Mathematics HSC Course

11B

8 In Geography Carlos scored a mark of 56, while in Business studies he scored 58. In Geography x = 64 and n = 10. For Business studies x = 66 and n = 15. a Convert each mark to a standardised score. b In which subject did Carlos achieve the better result?

11B

9 A psychologist records the number of errors made on a series of tests. On a literacy test the mean number of errors is 15.2 and the standard deviation is 4.3. On the numeracy test the mean number of errors is 11.7 with a standard deviation of 3.1. Barry does both tests and makes 11 errors on the literacy test and 8 errors on the numeracy test. In which test did Barry do better? Explain your answer.

11C

10 A data set is normally distributed with a mean of 40 and a standard deviation of 8. What percentage of scores will lie in the range: a 32 to 48? b 24 to 56? c 16 to 64?

11C

11 The value of sales made on weekdays at a store appears to be normally distributed with a mean of $1560 and a standard deviation of $115. On what percentage of days will the days’ sales lie between: a $1445 and $1675? b $1330 and $1790? c $1215 and $1905?

11C

12 A data set is normally distributed with a mean of 56 and a standard deviation of 8. What percentage of scores will: a lie between 56 and 64? b lie between 40 and 56? c be less than 40? d be greater than 80? e lie between 40 and 80?

11C

13 A machine is set to produce bolts with a mean diameter of 5 mm with a standard deviation of 0.1 mm. A bolt is chosen and it is found to have a diameter of 4.5 mm. What conclusion can be drawn about the settings of the machine?

Practice examination questions 1 multiple choice The mean time for 12-year-old boys to swim 50 m is 50.5 s with a standard deviation of 4.2 s. Kyle swims 50 m in 44.2 s. Kyle’s time as a standardised score is: A 6.3 B 1.5 C 1.5 D 6.3 2 multiple choice A teacher converts the marks on every test that she gives her class to a standardised score. On a test the mean mark was 50 and the standard deviation was 10. Adam’s standardised score on the test was 0.6. Adam’s mark on the test was: A 40 B 44 C 56 D 60

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3 multiple choice The details of Andrea’s half-yearly exams are shown in the table below. Subject

Andrea’s mark

Mean

Standard deviation

English

65

50

12

Maths

62

52

6

History

75

58

15

Geography

50

44

4

In which subject did Andrea achieve her best result? A English B Maths C History

D Geography

4 multiple choice The details of Brett’s half-yearly exams are shown in the table below. Subject

Brett’s mark

Mean

Standard deviation

English

40

50

12

Maths

48

52

6

History

49

58

15

Geography

42

44

4

In which subject did Brett achieve his best result? A English B Maths C History

D Geography

5 multiple choice A data set is normally distributed with x = 25 and lie in the range 25 to 30 is: A 34% B 47.5% C 68%

n

= 2.5. The percentage of scores that will D 95%

6 multiple choice A fishing boat catches a load of fish and finds the mass of each fish. The masses of the fish are normally distributed with a mean of 800 g and a standard deviation of 75 g. If a fish is chosen from the catch, its mass will almost certainly lie between: A 725 g and 875 g B 650 g and 950 g C 575 g and 1025 g D 800 g and 1025 g

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7 Theresa attempts to review her exam results in Physics and Chemistry. Theresa samples 10 of her friends and finds the following results. Physics: 65 64 67 69 72 50 66 66 63 69 Chemistry: 72 50 69 55 62 68 51 75 78 44 a Find the mean and standard deviation in each subject. b Theresa’s marks were 65 in Physics and 67 in Chemistry. Convert each to a standardised score. c In which subject did Theresa score her best result? Explain your answer. d A student is chosen at random from the Physics class. Between what two marks will this person’s result very probably lie? e If the marks within the class follow a normal distribution, within what two marks will approximately 99.7% of all Chemistry scores lie?

CHAPTER

test yourself

11

8 A machine is set to cut lengths of metal such that the mean length of metal cut is 12.5 cm with a standard deviation of 0.05 cm. a A piece of metal is measured to have a length of 12.4 cm. Express this as a standardised score. b A second piece of metal is measured and found to have a length of 13 cm. What conclusion can be drawn from this measurement?

Correlation

12 syllabus reference Data analysis 7 • Correlation

In this chapter 12A Scatterplots 12B Fitting a straight line by eye 12C Fitting a straight line — the 3-median method 12D Correlation

areyou

READY?

Are you ready?

Try the questions below. If you have difficulty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy.

12.1

12.2

Finding the median

1 Find the median of: a 3, 5, 6, 3, 4, 2, 5, 2, 7

Using the regression equation to make predictions

2 For the equation y = 5x – 2 find: a y if x = 40

12.3

b 12, 15, 10, 11, 15, 15, 16, 11, 19, 16.

b x if y = 258.

Finding the gradient I

3 Find the gradient of the line joining the points: a (1, 3) and (4, 12) b (–2, –4) and (6, –2).

12.4

Finding the gradient II

4 Calculate the gradient of the following lines, and state whether the gradient is positive or negative. a Vertical rise = 12, horizontal run = 2 b Vertical rise = –6, horizontal run = 4

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Scatterplots The manager of a small ski resort has a problem. He wants to be able to predict the number of skiers using his resort each weekend in advance, so that he can organise additional resort staffing and catering if needed. He knows that good deep snow will attract skiers in big numbers but scant covering is unlikely to attract a crowd. To investigate the situation further, he collects the following data over twelve consecutive weekends at his resort. Depth of snow (m)

Number of skiers

0.5

120

0.8

250

2.1

500

3.6

780

1.4

300

1.5

280

1.8

410

2.7

320

3.2

640

2.4

540

2.6

530

1.7

200

Number of skiers

As there are two types of data in this example, they are called bivariate data. For each item (weekend), two variables are considered (depth of snow and number of skiers). When analysing bivariate data, we are interested in examining the relationship between the two variables. In the case of the ski resort data we might be interested in answering the following questions. • Are visitor numbers related to depth of snow? 800 • If there is a relationship between visitor numbers and depth of snow, is it always true? or 600 is it just a guide? In other words, how strong is 400 the relationship? 200 • How much confidence could be placed in the 0 prediction? 0 1 2 3 4 To help answer these questions, the data can be Depth of snow (m) arranged on a scatterplot. Each of the data points is represented by a single visible point on the graph. When drawing a scatterplot, it is important to choose the correct variable to assign to each of the axes. The convention is to place the independent variable on the x-axis and the dependent variable on the y-axis. The independent variable in an experiment or investigation is the variable that is deliberately controlled or adjusted by the investigator. The dependent variable is the variable that responds to changes in the independent variable.

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Maths Quest General Mathematics HSC Course

Neither of the variables involved in the ski resort data was controlled directly by the investigator, but ‘Number of skiers’ would be considered the dependent variable because it is likely to change depending on depth of snow. (The snow depth does not depend on numbers of skiers). As ‘Number of skiers’ is the dependent variable, we graph it on the y-axis and the ‘Depth of snow’ on the x-axis. Notice how the scatterplot for the ski resort data shows a general upward trend. It is not a perfectly straight line, but it is still clear that a general trend or relationship has formed: as the depth of snow increases, so too does the number of skiers.

WORKED Example 1

The table below shows the height and mass of ten Year 12 students. Height (cm) Mass (kg)

120

124

130

135

142

148

160

164

170

175

45

50

54

59

60

65

70

78

75

80

Display the data on a scatterplot.

1 2

WRITE

Show the height on the horizontal axis and the mass on the vertical axis. Plot the point given by each pair.

Mass (kg)

THINK

80 70 60 50 40 30 0

100 110 120 130 140 150 160 170 180 Height (cm)

Graphics Calculator tip! Drawing a scatterplot Your graphics calculator can draw a scatterplot by storing the two sets of data in separate lists. Consider worked example 1. 1. From the MENU select STAT.

2. Delete any existing data, and store the data for height in List 1 and mass in List 2.

Chapter 12 Correlation

353

3. Press F1 (GRPH) (you may have to press F6 for more options first); then press F6 (SET). Set the graph type to Scatter by arrowing down to graph type and pressing F1 (Scat) (again you may have to press F6 for more options first). Ensure that XList is List 1, YList is List 2 and Frequency is 1 as shown at right. 4. Press EXIT to return to the previous screen, and then press F1 (GPH1). The scatterplot will then be drawn.

Note that the graphics calculator sets the values on the x- and y-axes automatically. You can press SHIFT F3 (V-Window) to set the scale as you see fit.

Once the scatterplot has been drawn, we can determine if any pattern is evident. Worked example 1 shows how, as a general rule, as height increases so does mass. We can also look to see if the pattern is linear. In worked example 1, although the points are not in a perfect straight line, they approximate a straight line. The figures below show examples of linear and non-linear relationships.

Linear relationships y

y

0

x

0

x

Non-linear relationships y

0

y

x

x

y

y

0

0

x

0

x

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Maths Quest General Mathematics HSC Course

In other cases it may be that there is no relationship at all between the two variables. Such a scatterplot would look like the one shown on the right.

y

0

x

WORKED Example 2

The table below shows the length and mass of a dozen eggs. Length (cm)

6.2

3.9

4.5

5.8

7.2

7.6

6.1

6.7

7.3

5.1

6.0

7.3

Mass (g)

60

15

25

50

95

110

55

75

95

35

54

96

a Display this information in a scatterplot. b Determine if there is any relationship between the length and mass of the eggs and state if the relationship is linear. THINK

WRITE

a

a

1

120 100 Mass (kg)

2

Display length on the x-axis and mass on the y-axis. Plot the point given by each pair.

80 60 40 20 0 0

b

1 2

Study the scatterplot to see if mass increases as length increases. Study the scatterplot to see if the points seem to approximate a straight line.

1

2

3

4 5 6 Length (cm)

7

8

b As length increases, so does the mass of the egg. The points do not approximate a straight line, and so the relationship is not linear.

remember 1. A scatterplot is a graph that is used to compare two variables. 2. One variable (the independent variable) is on the horizontal axis, and the other variable (the dependent variable) is on the vertical axis. 3. Points are plotted by the pair formed by each variable. 4. A relationship between the variables exists if one increases as the other increases or if one decreases as the other increases. 5. If the points on the scatterplot seem to approximate a straight line, the relationship can be said to be linear.

Chapter 12 Correlation

12A

355

Scatterplots E

65

82

72

58

39

58

74

82

66

Geography

45

78

66

72

50

51

61

70

60

88

L Spre XCE ad

E

36

Two variable 2 The table below shows the maximum temperature each day, together with the number statistics

of people who attend the cinema that day. Display the information on a scatterplot. Temperature (°C) No. at cinema

25

33

30

22

15

18

27

22

28

20

256

184

190

312

458

401

200

357

312

423

3 The table below shows the wages, W, of 20 people and the amount of money they spend each week on entertainment, E. Display this information in a scatterplot. Wages ($)

370

380

500

510

395

430

535

490

495

550

55

85

150

75

145

100

130

115

70

150

Wages ($)

810

460

475

520

530

475

610

780

350

460

Amount spent on entertainment ($)

220

50

100

150

140

160

90

130

40

50

Amount spent on entertainment ($)

WORKED

Example

2

4 The table below shows the marks obtained by nine students in English and History. English

55

20

27

33

73

18

37

51

79

History

72

37

53

74

73

44

59

55

84

a Display the information on a scatterplot. b Is there any relationship between the mark obtained in English and in History? If there does appear to be a relationship, is the relationship linear? 5 The table below shows the daily temperature and the number of hot pies sold at the school canteen. Temperature (°C)

24

32

28

23

16

14

26

20

29

21

No. of pies sold

56

20

24

60

84

120

70

95

36

63

a Display the information on a scatterplot. b Determine if there appears to be any relationship between the two variables and if the relationship appears to be linear.

sheet

History

sheet

L Spre XCE ad

1 The table below shows the marks obtained by a group of ten students in History and Example Scatterplot Geography. Display this information on a scatterplot. 1 WORKED

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Maths Quest General Mathematics HSC Course

6 Container ships arriving on a wharf are unloaded by work teams. The table below shows the number of people in the work team and the time taken to unload the container ship. No. in work team

15 18 12 19 22 21 17 16 18 20

Hours taken

20 16 25 15 14 13 18 20 17 14

a Display the information on a scatterplot. b Determine if there appears to be a relationship between the number of people in the work team and the time taken to unload the container ship. If there is a relationship, does the relationship appear to be linear? 7 multiple choice Which of the following scatterplots does not display a linear relationship? A y

B y

x x

C y

D y

x

x

8 multiple choice In which of the following is no relationship evident between the variables? A y

B y

x x

Cy

Dy

x

x

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357

9 Give an example of a situation where the scatterplot may look like the ones below. a y b y

0

x

0

x

Collecting bivariate data 1 Choose one of the following and collect data from within your class. a Each person’s hand span and height. b Each person’s resting heart rate and the time it takes for them to run 400 m. c Each person’s mark in Mathematics and in Science. 2 Display the results on a scatterplot. 3 Discuss any relationship that may be evident between the two variables.

Regression lines The process of ‘fitting’ straight lines to bivariate data enables us to analyse relationships between the data and possibly make predictions based on the given data set.

Fitting a straight line by eye Consider the set of bivariate data points shown at right. In this case the x-values could be heights of married women, while y-values could be the heights of their husbands. We wish to determine a linear relationship between these two random variables.

y

x

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Of course, there is no single straight line that would go through all the points, so we can only estimate such a line. Furthermore, the more closely the points appear to be on or near a straight line, the more confident we are that such a linear relationship may exist and the more accurate our fitted line should be. Consider the estimate, drawn by eye in the figure below. It is clear that most of the points are on or very close to this straight line. This line was easily drawn since the points are very much part of an apparent linear relationship. However, note that some points are below the line and some are above it. Furthermore, if x is the height of wives and y is the height of husbands, it seems that husbands are generally taller than their wives. y Regression analysis is concerned with finding these straight lines using various methods so that the number of points above and below the lines are balanced.

Method of fitting lines by eye

x

There should be an equal number of points above and below the line. For example, if there are 12 points in the data set, 6 should be above the line and 6 below it. This may appear logical or even obvious, but fitting by eye involves a considerable margin of error.

WORKED Example 3 Fit a straight line to the data in the figure using the equal-number-of-points method.

y

x

THINK 1 2

Note that the number of points (n) is 8. Fit a line where 4 points are above the line. Using a clear plastic ruler, try to fit the best line.

DRAW y

x 3

The first attempt has only 3 points below the line where there should be 4. Make refinements.

4

The second attempt is an improvement, but the line is too close to the points above it. Improve the position of the line until a better balance between upper and lower points is achieved.

y

x y

x

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remember To fit a straight line by eye, when using bivariate data, make sure there are an equal number of points above and below the fitted line.

12B

Fitting a straight line by eye

The questions below represent data collected by groups of students conducting different environmental projects. The students have to fit a straight line to their data sets. Note: For many of these questions your answers may differ somewhat from those in the back of the book. The answers are provided as a guide but there are likely to be individual differences when fitting straight lines by eye. Example

3

1 Fit a straight line to the data in the scatterplots using the equal-number-of-points method. a y

b y

x

d y

c y

x

e y

x

f y

x x

g y

x

h y

i y

Work

WORKED

x

x

x

Fitting a straight line — the 3-median method Fitting lines by eye is useful, but it is not the most accurate of methods. Greater accuracy is achieved through closer analysis of the data. Upon closer analysis it is possible to find the equation of a line of best fit of the form y = mx + c, where m is the gradient and c is the y-intercept. Several mathematical methods provide a line with a more accurate fit.

T SHEE

12.1

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One of these methods is called the 3-median method and involves the division of the data set into 3 groups, and the use of the 3 medians in these groups to determine a line of best fit. It is used when data show a linear relationship. It can even be used when the data contain outliers. The 3-median method is best described as a step-by-step method. Step 1. Plot the points on a scatter diagram. This is shown in figure 1. Step 2. Divide the points into 3 groups using vertical divisions (see figure 2). The number of points in a data set will not always be exactly divisible by 3. Thus, there will be three alternatives, as follows. (a) If the number of points is divisible by 3, divide them into 3 equal groups, for example, 3, 3, 3 or 7, 7, 7. (b) If there is 1 extra point, put the extra point in the middle group, for example, 3, 4, 3 or 7, 8, 7. (c) If there are 2 extra points, put 1 extra point in each of the outer groups, for example, 4, 3, 4 or 8, 7, 8. Step 3. Find the median point of each of the 3 groups and mark each median on the scatterplot (see figure 3). Recall that the median is the middle value. So the median point of each group has an x-coordinate that is the median of the x-values in the group and a y-coordinate that is the median of the y-values in the group. (a) The left group is the lower group and its median is denoted by (xL, yL). (b) The median of the middle group is denoted by (xM, yM). (c) The right group is the upper group and its median is denoted by (xU, yU). Note: Although the x-values are already in ascending order on the scatterplot, the y-values within each group may need reordering before you can find the median.

y 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x Figure 1 y 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x Figure 2 FM Fig 03.13 y 7 6 5 4 3 2 1 0

(xU, yU) (xM, yM) (xL, yL)

0 1 2 3 4 5 6 7 8 x Figure 3 y 7 6 5 4 3 2 1 0

(xM, yM)

(xU, yU)

(xL, yL) Step 4. Draw in the line of best fit. Place your ruler so that it passes through the lower and upper medians. Move the ruler a third of the way toward the middle group 0 1 2 3 4 5 6 7 8 x median while maintaining the slope. Hold the ruler Figure 4 there and draw the line. Step 5. Find the equation of the 3-median regression line (general form y = mx + c). Draw on your knowledge of finding equations of lines to find the equation of the line drawn on the scatterplot. If the scale on the axes begins at zero, you can read off the y-intercept of the line and calculate the gradient of the line. This will enable you to find the equation of the line.

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The equation of a straight line can be found using y = mx + b , where m is the gradient and b is the y-intercept. The gradient of the regression line is best found with a ruler and using the formula: vertical change in position m = --------------------------------------------------------------------horizontal change in position

WORKED Example 4

Find the equation of the regression line for the data in the table at right using the 3-median method. THINK 1

x

1

2

3

4

5

7

y

1

3

2

6

5

6

WRITE

Plot the points on a scatterplot, and divide the data into 3 groups. Note there are 6 points, so the division will be 2, 2, 2.

y 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x

2

3

Find the median point of each group. Since each group has only 2 points, medians are found by averaging them. Mark in the medians, and place a ruler on the outer 2 medians. Maintaining the same slope on the ruler, move it onethird of the way towards the middle median. Draw the line.

(xL, yL) = (1.5, 2) (xM, yM) = (3.5, 4) (xU, yU) = (6, 5.5) y 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x

4

Read off the y-intercept from the graph.

5

Use (xL, yL) and (xU, yU) to calculate the gradient.

6

Write the equation of the 3-median regression line.

y-intercept = 1 5.5 – 2 Gradient (m) = ---------------6 – 1.5 3.5 = ------4.5 7 = --9 7 y = --- x + 9 or 9 9y = 7x + 81

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WORKED Example 5

Weight (kg)

The scatterplot below shows a comparison between the heights and weights of 12 boys. The median points A and B in the first and last sections have been found for you.

100 90 80 70 60 50 40 30 20 10

A

0

B

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 Height (cm)

a Find the coordinates of median point C, and hence find the median regression line. b Find the gradient and y-intercept of the regression line, and hence find the equation of the regression line. THINK

WRITE

a

a x-values are: 165, 170, 170 and 175 170 + 170 Median x-value = ------------------------ = 170 2 y-values are: 65, 70, 70 and 80 70 + 70 Median y-value = ------------------ = 70 2 The coordinates of C are (170, 70).

Find the coordinates of point C by finding the median of the x-values and finding the median of the yvalues.

2

Mark point C on the diagram.

3

Rule a line through points A and B.

4

Move the line AB one-third of the way towards C, keeping the new line parallel to AB.

Weight (kg)

1

100 90 80 70 60 50 40 30 20 10 0

A C

B

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 Height (cm)

Chapter 12 Correlation

THINK

WRITE

b

rise b m = -------run 78 – 70 m = -----------------------190 – 140 m = 0.16 b = 49

1

Calculate the gradient, m, by finding the rise and run between two points on the line.

2

Read the value from the graph to state the y-intercept, b. Substitute m and b into the formula y = mx + b to find the equation of the regression line.

3

363

The equation is of the form y = mx + b, where x represents height in cm and y represents weight in kg. y = 0.16x + 49

the equation of Graphics Calculator tip! Finding a regression line The Casio graphics calculator can be used to find the equation of a median regression line. Consider worked example 5. 1. From the MENU select STAT.

2. Enter the data into List 1 and List 2 and draw the scatterplot as shown in the previous section. Since we are using the calculator it is not necessary to draw the scatterplot from 0 on the axes. 3. Press F2 (Med) to find the equation of the median regression line. The value of a is the gradient of the line and the value of b is the y-intercept.

4. If you want to see the regression line drawn on the scatterplot, press F6 (DRAW).

In the above example we would give the equation y = 0.15x + 49, which is slightly different from the example done on paper. Because the method relies on the eye to find two points on the regression line to find the gradient and y-intercept, minor differences are insignificant and quite acceptable. Once the regression line has been found, we are able to use the equation to make predictions about other pieces of data.

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WORKED Example 6 A casino records the number of people, N, playing a jackpot game and the prize money, p, for that game and plots the results on a scatterplot. The regression line is found to have the equation N = 0.07p + 220. a Find the number of people playing when the prize money is $2500. b Find the likely prize on offer when there are 500 people playing. THINK

WRITE

a

a N = 0.07p + 220

b

1

Write the equation of the regression line.

2

Substitute 2500 for p.

3

Calculate N.

4

Give a written answer.

1

Write the equation of the regression line.

2

Substitute 500 for N.

500 = 0.07p + 220

3

Solve the equation.

280 = 0.07p p = 4000

4

Give a written answer.

The prize would be approximately $4000.

N = 0.07

2500 + 220

= 395 There would be approximately 395 people playing. b

N = 0.07p + 220

remember 1. The median regression line is the line of best fit that is drawn on a scatterplot. 2. The median regression line can be drawn using the method of three medians. 3. To find the median regression line: (a) divide the points into three approximately equal sections. If the number of points is not divisible by three, make sure there is the same number of points in the first and last sections. (b) mark median points in the first and last sections by finding the median of the x-values and finding the median of the y-values for each section. Label these points A and B. (c) find the median point in the middle section and label this point C. (d) draw the line AB and then move the line one-third of the way towards C, keeping the line parallel to AB. 4. The equation of the regression line can be found by measuring the gradient and the y-intercept of the regression line and using the formula y = mx + b . Sometimes the gradient of the median regression line will be negative. 5. Once the equation of the regression line has been found, it can then be used to make predictions about the variables.

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12C WORKED

Example

Fitting a straight line — the 3-median method

1 The table below shows the marks achieved by a class of students in English and Maths. English

64

75

81

63

32

56

47

59

73

64

Maths

76

62

89

56

49

57

53

72

80

50

12.1 SkillS HEET

4

Show these data on a scatterplot, and on the graph show the regression line using the 3-median method.

Finding the median

12.2 SkillS

60 50 40 30 20 10 0 0

2

4

6

8

10

to make predictions

12.3 SkillS HEET

60 50 40 30 20 10 0

HEET

2 Position the median regression line, using the 3-median method, through each of the Using following graphs, and find the equation of each. the regression a 70 b 70 equation

0

20 40 60 80 100 120

c

Finding the gradient I

3000

12.4 SkillS

2500

HEET

2000

Finding the gradient II

1500 1000

E

0 0 Example

5

3-median regression

10 15 20 25

3 In an experiment, a student measures the length of a spring when different masses are attached to it. Her results are shown below. Mass (g) 0 100 200 300 400 500 600 700 800 900

Length of spring (mm) 220 225 231 235 242 246 250 254 259 264

Making predictions

a Draw a scatterplot of the data, and on it draw the median line of regression, using the 3-median method. b Find the gradient and y-intercept of the regression line, and hence find the equation of the regression line.

sheet

L Spre XCE ad

E

WORKED

5

sheet

L Spre XCE ad

500

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4 A scientist who measures the volume of a gas at different temperatures provides the table of values at right. a Draw a scatterplot of the data and on it draw the line of regression using the 3-median method. b Give the equation of the line of best fit. Write your equation in terms of the variables: volume of gas, V, and its temperature, T. 5 A sports scientist is interested in the importance of muscle bulk to strength. He measures the biceps circumference of ten people and tests their strength by asking them to complete a lift test. His results are given in the following table. Circumference of biceps (cm) 25 25 27 28 30 30 31 33 34 36

Temperature (°C) 40 30 20 0 10 20 30 40 50 60

Volume (L) 1.2 1.9 2.4 3.1 3.6 4.1 4.8 5.3 6.1 6.7

Lift test (kg) 50 52 58 51 60 62 53 62 61 66

a Draw a scatterplot of the data and draw the median line of regression using the 3-median method. b Find a rule for determining the ability of a person to complete a lift test, S, from the circumference of their biceps, B. WORKED

Example

6

6 A taxi company adjusts its meters so that the fare is charged according to the following equation: F = 1.2d + 3, where F is the fare, in dollars, and d is the distance travelled, in km. a Find the fare charged for a distance of 12 km. b Find the fare charged for a distance of 4.5 km. c Find the distance that could be covered on a fare of $27. d Find the distance that could be covered on a fare of $13.20. 7 Detectives can use the equation H = 6.1f 5 to estimate the height of a burglar who leaves footprints behind. (H is the height of the burglar, in cm, and f is the length of the footprint.) a Find the height of a burglar whose footprint is 27 cm in length. b Find the height of a burglar whose footprint is 30 cm in length. c Find the footprint length of a burglar of height 185 cm. (Give your answer correct to 2 decimal places.) d Find the footprint length of a burglar of height 152 cm. (Give your answer correct to 2 decimal places.)

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8 A pie seller at a football match finds that the number of pies sold is related to the temperature of the day. The situation could be modelled by the equation N = 870 23t, where N is the number of pies sold and t is the temperature of the day. a Find the number of pies sold if the temperature was 5 degrees. b Find the number of pies sold if the temperature was 25 degrees. c Find the likely temperature if 400 pies were sold. d How hot would the day have to be before the pie seller sold no pies at all? 9 The following table shows the average annual costs of running a car. It includes all fixed costs (registration, insurance etc.) as well as running costs (petrol, repairs etc.). Distance (km)

Annual cost ($)

5 000

4 000

10 000

6 400

15 000

8 400

20 000

10 400

25 000

12 400

30 000

14 400

a Draw a scatterplot of the data. b Using the 3-median method, draw in the line of best fit. c Find an equation which represents the relationship between the cost of running a vehicle, C, and the distance travelled, d. d Use your graph and its equation to find: i the annual cost of running a car if it is driven 15 000 km ii the annual cost of running a car if it is driven 1000 km iii the likely number of kilometres driven if the annual costs were $8000 iv the likely number of kilometres driven if the annual costs were $16 000. 10 A market researcher finds that the number of people who would purchase ‘Wise-up’ (the thinking man’s deodorant) is related to its price. He provides the table of values at right. a Draw a scatterplot of the data. b Draw in the line of best fit. c Find an equation that represents the relationship between the number of cans of ‘Wise-up’ sold, N (in thousands), and its price, p. d Use the equation to predict the number of cans sold each week if: ii the price was $3.10 ii the price was $4.60. e At what price should ‘Wise-up’ be sold if the manufacturers wished to sell 80 000 cans? f Given that the manufacturers of ‘Wise-up’ can produce only 100 000 cans each week, at what price should it be sold to maximise production?

Price ($)

Weekly sales ( 1000)

1.40

105

1.60

101

1.80

97

2.00

93

2.20

89

2.40

85

2.60

81

2.80

77

3.00

73

3.20

69

3.40

65

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11 The following table gives the adult return air fares between some Australian cities. City Melbourne–Sydney Perth–Melbourne Adelaide–Sydney Brisbane–Melbourne Hobart–Melbourne Hobart–Adelaide Adelaide–Melbourne

Distance (km) 713 2728 1172 1370 559 1144 669

Price ($) 580 1490 790 890 520 820 570

a Draw a scatterplot of the data and on it draw the median regression line using the line of best fit. b Find an equation that represents the relationship between the air fare, A, and the distance travelled, d. c Use the equation to predict the likely air fare (to the nearest dollar) from: i Sydney to the Gold Coast (671 km) ii Perth to Adelaide (2125 km) iii Hobart to Sydney (1024 km) iv Perth to Sydney (3295 km). 12 Rock lobsters (crayfish) are sized according to the length of their carapace (main body shell). The table below gives the age and carapace length of 16 male rock lobsters. Age (years) 3 2.5 4.5 4.5 3.25 7.75 8 6.5 12 14 4.5 3.5 2.25 1.76 10 9.5

Length of carapace (mm) 65 59 80 80 68 130 150 112 200 210 82 74 51 48 171 160

a Display this information on a scatterplot, and on your scatterplot draw the median line of regression using the line of best fit. b Find the equation of the median regression line.

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c Use the equation to find the likely size of a 5-year-old male rock lobster. d Use the equation to find the likely size of a 16-year-old male rock lobster. e Rock lobsters reach sexual maturity when their carapace length is approximately 65 mm. Use the equation to find the age of the rock lobster at this stage. f The fisheries department wants to set minimum size restrictions so that the rock lobsters have three full years from the time of sexual maturity in which to breed before they can be legally caught. What size should govern the taking of a male rock lobster? Note: Answers for this exercise are approximate and may vary due to the precise location of the line of best fit.

Relationship between variables Earlier in the chapter you would have completed an investigation of bivariate data. You should have displayed the information on a scatterplot. 1 On your scatterplot draw the median regression line. 2 Find the equation of the median regression line. 3 Find a few more people to test your data. See how accurately your equation predicts the results. (For example measure a person’s hand span and use your equation to predict their height.)

1 An electrical repair business charges its customers using the formula C = 40h + 35, where C is the cost of the repairs and h is the time taken for the repairs, in hours. Find the cost of a repair job that took: 1 2 hours

2 5 hours

3 1 hour and 15 minutes.

Estimate the time taken for repairs if the cost of the repairs were: 4 $175

5 $275

6 $145.

The information below is to be used for questions 7 to 10. A survey relating exam marks to the amount of television watched finds that the median regression line has the equation M = 95 15t, where M is the mark obtained and t is the average number of hours of television watched each night by the students. 7 Estimate the mark of a person who averages one hour of television per night. 8 Estimate the mark of a person who watches an average 4 hours of television per night. 9 Estimate the amount of television watched per night by a person who scores a mark of 65. 10 Jodie scored 27.5 on the exam. Estimate the average amount of television that Jodie watches each night.

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Correlation Correlation is a description of the relationship that exists between two variables. When one variable increases with another, it is said that there is a positive correlation between the variables. In such a case, the median regression line will have a positive gradient. Similarly, if one variable decreases while the other increases, the median regression line will have a negative gradient and the correlation is negative. Consider the following example in which ten Year 11 students were surveyed to find the amount of time that they spend doing exercise each week. This was compared with their blood cholesterol level. 6

8

12

16

2

0

5

8

7

12

Blood cholesterol level

4

3

3

3

9

8

9

6

5

4

In this example there seems to be a general downward trend, and the median regression line therefore has a negative gradient. As the amount of exercise increases, the level of blood cholesterol decreases. Notice that in this case the points are not as closely aligned as in the previous examples. We can say that the relationship (or correlation) between the variables is only weak. In general terms, the closer that the points are to forming a straight line, the stronger the relationship is between the variables. Sometimes we find that there is no relationship between the variables. In the scatterplot below, a researcher was looking for a link between people’s heights and their IQs. The points appear to be randomly dispersed across the scatterplot. In cases like this, it can be concluded that there is no clear relationship between the variables. 140 IQ

120 100 80 60 120 140 160 180 200 Height (m)

Blood cholesterol level

Period of exercise (h)

10 8 6 4 2 0 0

2

4

6 8 10 12 14 16 Period of exercise

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WORKED Example 7

In the figure on the right, describe the correlation as being positive or negative.

THINK

WRITE

1

Add a median regression line to the scatterplot.

2

The gradient of the regression line is positive. Therefore the correlation is positive.

3

There is a positive correlation.

The strength of a correlation is based on the correlation coefficient. The correlation coefficient is a measure of a correlation. Correlation coefficient 1

Description Perfect positive correlation

Between 0.75 and 1

Strong positive correlation

Between 0.5 and 0.75

Moderate positive correlation

Between 0.25 and 0.5

Weak positive correlation

Between −0.25 and 0.25

No correlation

Between −0.5 and −0.25

Weak negative correlation

Scatterplot

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Correlation coefficient Between −0.75 and −0.5

Description Moderate negative correlation

Between −1 and −0.75

Strong negative correlation

−1

Perfect negative correlation

Scatterplot

WORKED Example 8 The operators of a casino keep records of the number of people playing a ‘Jackpot’ type game and compare the numbers playing to the size of the jackpot. The correlation coefficient for this game is calculated to be 0.65. Describe the correlation between the prize and the number of players. THINK

WRITE

The correlation coefficient is between 0.5 and 0.75 and so it is a moderate positive correlation.

There is a moderate positive correlation between the jackpot and the number of players in the game.

Causality Causality refers to one variable causing another. For example, there is a high correlation between a person’s shoe size and shirt size. However, one does not cause the other. Similarly, there is a high correlation between number of cigarettes smoked and lung cancer but, in this case, smoking causes lung cancer. Explain whether a positive or negative relationship exists and discuss causality in each of the following. 1. Hours of study and exam marks 2. Hours of exercise and resting pulse rate 3. Weight and shirt size 4. The number of hotels and churches in country towns 5. The number of motels in a town and the number of flights landing at the nearest airport

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It is possible to make a qualitative judgement as to the type of correlation that is involved in a relationship by the general appearance of the graph. Care must be taken before making a statement about one variable causing the other. Just because there is a strong relationship between two variables, it does not mean that one variable causes the other. For example, there is a very strong positive correlation in people between their shoe size and their shirt size, but one does not cause the other. Similarly, there is a very strong correlation between the amount of study done for an exam and the result achieved on the exam. In this case it can be argued that the study causes the high exam mark. Each case needs to be considered on its merit.

WORKED Example 9

A manufacturer who is interested in minimising the cost of training gives 15 of his plant operators different amounts of training and then measures the number of errors made by each of the operators. The results of the experiment are placed on a scatterplot and the correlation between the number of hours of training and the number of errors made is measured to have a correlation coefficient of −0.69. a What can be said of the correlation between training and errors? b What conclusion could the manufacturer make about causality in this case? THINK

WRITE

a

a

The correlation coefficient is between −0.75 and −0.5. 2 A correlation coefficient in this range indicates a moderate negative correlation. b In this case it would seem logical that those that have undertaken more training would make fewer errors. 1

There is a moderate negative correlation between the amount of training and the number of errors made. b The manufacturer could reasonably presume that the more training a person is given, the less likely they are to make errors with the machinery.

remember 1. The pattern of the scatterplot gives an indication of the level of association (correlation) between the variables. 2. When one variable increases with another, there is a positive correlation between them. 3. When one variable decreases while the other increases, there is negative correlation. 4. The extent of the correlation is then measured by the correlation coefficient. The description of the correlation is given in the figure on the right. 5. Strong correlation between two variables does not necessarily mean that one variable causes the other.

Strong positive correlation Moderate positive correlation Weak positive correlation No correlation Weak negative correlation Moderate negative correlation Strong negative correlation

1

Perfect positive correlation

0.75 0.5 0.25 0 – 0.25 – 0.5 – 0.75 –1

Perfect negative correlation

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12D WORKED

Example

7

Correlation

1 For each of the following, state whether a positive or negative correlation exists. a

b

c

2 A sample of 10 drivers was taken. Each driver was asked their age and the number of speeding offences they had committed in the past five years. The results are in the table below. Age Speeding offences

22

36

48

40

58

64

23

25

30

45

4

2

1

1

2

0

3

7

1

0

a Display the information on a scatterplot. b State if there is a positive or a negative correlation between age and speeding offences. 3 Match each of the following scatterplots with the correlation that it shows. a

b

A

d

c

e

D

C

B

f

E

F

g

G

Strong positive correlation Weak positive correlation Weak negative correlation Strong negative correlation

Moderate positive correlation No correlation Moderate negative correlation

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4 A pie seller at a football match notices that there seems to be a relationship between the number of pies that he sells and the temperature of the day. He collects the following data. Daily temperature (°C)

12

Number of pies sold

22

26

11

8

18

14

16

15

16

620 315 295 632 660 487 512 530 546 492

a Draw a scatterplot of the data. b State the type of correlation that the scatterplot shows and draw a conclusion from the graph. 5 A researcher is investigating the effect of living in airconditioned buildings upon general health. She records the following data. Hours spent each week in airconditioned buildings

2

13

6

48 40

0

10

0

2

5

Number of days sick due to flu and colds

3

6

2

15 13

8

14

1

16

9

18 10

9

6

a Plot the data on a scatterplot. b State the type of correlation the graph shows and draw a conclusion from it. c The researcher finishes her experimental report by concluding that airconditioning is the cause of poor health. Is she correct to say this? What other factors could have influenced the relationship shown by the scatterplot? 6 The data below show the population and area of the Australian states and territories. State

Area (× 1000 km2)

Population (× 1000)

Vic.

228

5092

NSW

802

6828

ACT

2

329

Qld

1727

4053

NT

1346

207

WA

2526

2051

SA

984

1555

Tas.

68

489

a Plot the data on a scatterplot. b State the type of correlation the graph shows and draw a conclusion from it.

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7 In an experiment, 12 people were administered different doses of a drug. When the drug had taken effect, the time taken for each person to react to a set stimulus was measured. The results are detailed below. Amount of drug (mg)

Reaction time (s)

0.1

0.030

0.2

0.025

0.3

0.028

0.4

0.036

0.5

0.040

0.6

0.052

0.7

0.046

0.8

0.068

0.9

0.085

1.0

0.092

1.1

0.084

1.2

0.096

a Plot the data on a scatterplot. b State the type of correlation the graph shows, and draw a conclusion from it. 8 multiple choice What type of correlation is shown by the graph on the right? A No correlation B Weak negative correlation C Moderate negative correlation D Strong negative correlation 9 multiple choice What type of correlation is shown by the graph on the right? A No correlation B Weak positive correlation C Moderate positive correlation D Strong positive correlation 10 What type of correlation would be represented by scatterplots that had the following correlation coefficients? a 1.0 b 0.4 c 0.8 d −0.7 e 0.35 f 0.21 g −0.75 h −0.50 i −0.25 j −1.0

Chapter 12 Correlation

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11 A researcher investigating the proposition that ‘tall mothers have tall sons’ measures the heights of 12 mothers and the heights of their adult sons. The correlation coefficient 8 is found to be 0.67. Describe the correlation between tall mothers and tall sons.

WORKED

Example

12 A teacher who is interested in the amount of time students spend doing homework asks 15 students to record the amount of time that they spend on homework and on watching television. The correlation coefficient is found to be −0.45. Interpret the correlation between homework and television watching. 13 A psychologist asked 20 people to rate their ‘level of contentment’ on a scale of 0 to 10 (10 representing ‘perfectly content’). This rating is compared to annual income. a The correlation coefficient is found to be −0.18. Describe the correlation between income and level of contentment. b The researcher then intends to write an essay entitled ‘Money can’t buy happiness’. Do the results confirm this statement? 14 An experimenter who is investigating the relationship between exercise and obesity 9 measures the weights of 30 boys (of equal height) and also documents the amount of physical exercise that the boys completed each week. The correlation coefficient is found to be −0.47. a What can be said of the correlation between obesity and exercise? b What conclusion could be made about causality in this case?

WORKED

Example

A researcher is interested in the association between the work rate of production workers and the level of incentive that they are offered under a certain scheme. After drawing a scatterplot, she calculates the correlation between the two variables at 0.82. The researcher can conclude that: A There is a strong positive correlation between the variables; the greater the incentive, the lower the work rate. B There is a strong positive correlation between the variables; the greater the incentive, the greater the work rate. C There is a strong negative correlation between the variables; the greater the incentive, the lower the work rate. D There is a strong negative correlation; incentives cause an increase in the work rate.

Work

15 multiple choice

T SHEE

12.2

378

Maths Quest General Mathematics HSC Course

summary Scatterplots • When looking for a relationship between two variables, data can be represented on a scatterplot. • One variable (the independent variable) is on the x-axis and the other variable (the dependent variable) is on the y-axis. • Points are plotted by the coordinates formed by each piece of data. • If the dependent variable consistently increases or decreases as the independent variable increases, a relationship exists. • If all points on the scatterplot form a straight line, the relationship is said to be linear. • The pattern of the scatterplot gives an indication of the strength of the relationship or level of association between the variables. This level of association is called correlation. • A strong correlation between variables does not imply that one variable causes the other to occur.

Median regression lines • A regression line is the line of best fit on a scatterplot. • By measuring the gradient and the y-intercept on the regression line, we can use the formula y = mx + b to find the equation. • When the equation of a regression line has been found, it can then be used to make predictions about the data. • We can find the regression line by using the eye method or the method of 3-medians.

Correlation • Correlation is the measure of the relationship between two variables. • A correlation can be positive or negative and has the same sign as the gradient of the median regression line. • A positive correlation means that one quantity will increase as the other increases. • A negative correlation means that one quantity will decrease as the other increases. • Correlation can be quantified by using a correlation coefficient. • The correlation coefficient may be interpreted as follows: q=1 Perfect positive correlation 0.75 q < 1 Strong positive correlation 0.5 q < 0.75 Moderate positive correlation 0.25 q < 0.5 Weak positive correlation 0.25 < q < 0.25 No correlation 0.5 < q 0.25 Weak negative correlation 0.75 < q 0.5 Moderate negative correlation 1