Modeling A Functional Building IA

 

 

Modeling a Functional Building

An Internal Assessment for the International Baccalaureate Subject of Math HL

Candidate Name: Georgiy Kachurin CandidateHillcrest SessionHigh Number: 001395-070 School Midvale, Utah, USA School Code: 001395

Instructor: Kenneth Herlin Submission date: 21 January 2011 Examination Session: May 2011 Word Count: 1,268

 

A building was to be designed with the following specifications: The building has a rectangular base 150 m long and 72 m wide. The maximum height of the structure should not exceed 75% (i.e. 54 m) of its width for stability or be less than half the width for aesthetic purposes. The minimum height of a room in a public building is 2.5 m. A model for a curved roof structure was to be created when the height of the building is 36 m. The standard parabolic equation for such a roof structure would look as follows:   In this situation, would be 36 m. is yet to be calculated. If this structure would be imagined on a standard Cartesian graph, the curve would begin and end at the points (-36, 0) and (36, 0), respectively. Thus, is solved for by setting and (because is being squared, it does not matter if 36 or –  or – 36 36 are substituted for ) and simplifying as follows:    





  



      ()                Thus, the model for the curved roof structure is:          



Next, a cuboid with maximum volume needs to be inscribed inside this roof structure. This is an optimization problem. Because the height of the roof structure does not depend on the length of the base (150 m), but rather the width (72 m), the length can be neglected for now (as long as it is factored back in at the end of the problem). Assuming that ( is the point on the parabolic curve that meets the corner of the rectangle, and that the rectangle is symmetrical about the axis, the width of the rectangle can be signified by . The height of the rectangle is

)





      . Because the area of a rectangle is found by , the area of this rectangle is:     Simplified:    (  )        

signified by

Since

  is differentiable, and the only critical points occur at the zeros of the first derivative:  )   (                        

 

       By substituting this value of  into  =  and       , the values 41.57 and 24 are  discovered, respectively. However, since the volume of a cuboid is needed, the value of      needs to be factored in. Thus, the cuboid has dimensions:   x   x  , amounting a maximum volume . of the structure affect the It isto interesting to note howof  the changes to the height           dimensions of the largest possible cuboid. For example, if the building had a maximum height of   , the parabolic curve would have been solved likewise and would come out as:          Thus, the dimensions of the rectangle would be: (   ). When the derivative of both  sides has been taken, the result set equal to zero, solved for  and substituted, the values come out to     and    . A peculiar fact was that when the  value was obtained, it was the same as the  value obtained from the curve of minimum height (). It is due to this fact that the width of the rectangle has remained unchanged. THIS ALSO HOLDS TRUE  . FOR ANY OTHER HEIGHT: the width will always be         The volume of the building is calculated by taking the integral of the curve (thus giving the area under the curve), and then multiplying that by :

 

  ∫       

 

By utilizing a TI-84 Plus Silver Edition Calculator, this integral was found to equal to:   By multiplying this value by , the volume is equal to:   The volume of wasted space is found by subtracting the volume of the office block from the total volume of the building: . Thus, the ratio of the volume of wasted space to the volume of the office block for a height of  is:     Using these same steps, the ratio of the volume of wasted space to the volume of the office block  for a height of  is:

                 

 

 

       ∫                                                               

 

These results seem to show that the greater the height of the roof structure, the more wasted space there is, most likely due to the fact that area underneath the parabolic curve is increasing, but the width of the rectangle is staying the same. The maximum floor area needs to be determined, as well. For the roof structure of height , the height of the cuboid is . Because the minimum height of a room in a public building is , there are 9.6 floors in this building. Because each floor has a base of  . The , the amount of floor space is building of maximum height has floors ( ), and thus has of floor space. If the façade is placed on the longer side of the base, meaning that the width is now considered to be instead of  . Here, the minimum height would be , and the maximum would be . The standard parabolic curve here would look like:  

 

   

                                           Where       . The minimum height parabolic curve would be equal to:            ∫                    Compared to the other building’s volume of    , this is approximately 4 times greater, and this is just the minimum height! This is probably due to the fact that more height is allowed because of the increased width of the building. Now for the maximum height:

      ∫                         Again, compared to the other building’s volume of    , this volume is significantly larger (more than 3 times larger).

Fig. 1 –  1 – the the graphs of the curved roof structures with their corresponding rectang (theside larger curve is when the façadeinscribed is placed rectangles on the les longer of the base)

 

In order to maximize office space, the block must not be in the shape of a single cuboid. In order to do so, a Riemann sum may be calculated, sampled at the minimum (the roof structure is treated as a boundary line; the cuboid must not exceed this boundary) using 12 subintervals.

       (the height of the roof =  ) was found to be  . Thus, the volume is           . Likewise, the volume for       (the height of the roof  ) was          . Using technology, the sum for



Figs. 2 & 3 –  3 – graphs graphs of the curved roof structures (same width, different heights) with their corresponding Riemann sums, using 12 rectangles.

In order to find the amount of wasted space after the Riemann sum, it is necessary to subtract the estimated volume from the actual volume:   Height = : Height = :  

                              

Table 1: Floor area, before and after  Height of roof  Floor area (before) structure        

   

Floor area (after)

Percent Increase (%)

   

Table 2: Ratios of the volume of wasted space to office block space, before and after 

Height of roof  structure    

   

Ratio to the volume of wasted space to office block space (before)

          

Ratio to the volume of wasted space to office block space (after)    

              

Percent Increase (%) 481.606% 257.020%