Differential Equation - Problem Set.docx

1. Radium decompose at the rate proportional to the amount present. It is found out that in 25 years, 1.1% of a certain amount decomposed. Determine approximately how long will it take a. for one half the original amount to decompose? b. 20% of the original amount to decompose? c. What % will decompose in 50 years? 2. A bacterial population is known to have a growth rate proportional to B itself. If between noon and 2 PM, the population triples, at what time, no controls being exerted should B becomes 100 times what is was at noon? What will be the population at 6 PM? 3. The population of the town grows at the rate proportional to the population present at any time t. The initial population of 500 increases by 15% in 10 years. What will be the population in 30 years? 4. A metal is heated up to a temperature of 500 deg C. It is then exposed to a temperature of 38 deg C. After 2 min, the temperature of the metal becomes 190 deg C. When will the temperature be 100 deg C? What is the temperature after 4 minutes? 5. A thermometer is taken from an inside room to the outside, where the air temperature is 5 deg F. After 1 min, the thermometer reads 55 deg F and after 5 min, it reads 30 deg F. What was the initial temperature of the oven? 6. A thermometer is removed from a room where the air temperature is 70 deg F and is taken outside where air temperature is 10 deg F. After ½ min, the thermometer reads 50 deg F. What is the reading of the thermometer at t = 1 min? How long will it take for the thermometer to reach 15 deg F? 7. A tank contains 100 gal of brine with 5 lbs of dissolved salt. Brine enters a tank at the rate of 3 gal per min with 2 lb of salt per gal. The solution well stirred leaves at the same rate. Find the amount of salt in the tank at any time. What is the amount of salt after 7 min? 8. A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4 liters per minute, the well mixed solution is pumped out at the same rate. Find the number of grams X (t) of salt in the tank at any time. 9. When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I (t) where t represents the thickness of the medium in feet(ft). In clear sea water, the intensity 3 feet below the surface is 25% of the initial intensity I0 of the incident beam. What is the intensity of the beam 15 feet below the surface? 10. For a substance X, the time rate of conversion is proportional to the square of the amount x of unconverted substance. Let k be a numerical value of the constant of proportionality and x0 be the amount of unconverted substance at time = 0. Find x at any time t.

11. The population of a certain country doubles in 50 years. When will be the population be tripled? Assume law of growth to hold. 12. The population growth of a certain colony of mosquitoes follows the uninhabited growth equation. If there are 1,500 mosquitoes initially, and there are 2,500 mosquitoes after 24 hours, what is the size of the mosquito population after 3 days? 13. If the half-life of radium is 1,700 years, what percentage radium may be expected to remain after 50, 100, and 200 years? 14. The rate of decay of a radioactive substance is proportional to the amount present. If half of a given deposit of a substance disappears in 1,455 years, how long will it take for 25% of the deposit to disappear? 15. A piece of charcoal is only 11% or 14C of its original amount and the half-life of 14C is 2,300 years. When was the tree cut from which the charcoal came from? 16. A radioactive substance decreases from 10 g to 9 g in one hour. Find its half-life. 17. In a chemical transformation, substance A changes into another substance at a rate proportional to the amount of A unchanged. If initially there was 40 grams of A and one hour later there was 12 grams left, when will 90% of A be transformed? 18. In a culture of yeast, the amount of active ferment grows at a rate proportional to the amount present. If the culture doubles in one hour, how many times the original amount may be anticipated at the end of 2.75 hours? 19. If 30% of the radioactive substance disappears in 10 days, how-long will it take for 90% to disappear? 20. The amount of radioactive isotope C14 present in all living organic matter bears a constant ratio to the amount of the stable to isotope C12. An analysis of fossil remains of dinosaur shows that the ratio is only 6.24% of that living matter. Assuming the half-life of C14 is approximately 5,600 years; determine how long ago the dinosaur was alive. 21. The population of a suburb doubled in size in an 18-month period. If this growth continues and the current population is 8,000, what will the population be in 4 years? 22. At any time t, the rate of increase in the area of a culture of bacteria is twice the area of the culture. If the initial area of the culture is 10, then what is the area at time t?

23. A substance decomposes at the rate proportional to the quantity of substance present. If in 36 years only 2.3% of it has decomposed, determine the half-life. 24. A paleontologist discovered an insect preserved inside a transparent amber, which happened to be a tree pitch, and the amount 14C present in the insect was determined to be 20% of its original amount. Use the fact that half-life of 14C is 59,000 years to determine the age of the insect at the time of discovery. 25. Bacteria in a certain culture increases at a rate proportional to the number present. If the original number increases by 50% in ½ hour, in how many hours can one expect three times the original number and five times the original number? 26. Radium decomposes at a rate proportional to the amount present, if the half-life is 1,600 years, that is, if half of any given amount is decomposed in 1,600 years, find the percentage remaining at the end of 200 years. 27. Find the half-life of a radioactive substance if 25% of it disappears in 10 years. 28. Find the time required for a sum of money to double itself at 5% per annum compounded continuously. 29. A certain radioactive substance has a half-life of 38 hours. Find how long it takes for 90% of the radioactivity to be dissipated. 30. If the thermometer is taken from a room in which the temperature is 75o into the open, where the temperature is 35o and the reading of the thermometer was 65o after 30 seconds. a. How long after the removal will the reading be 55o? b. What is the thermometer reading 3 minutes after the removal? 31. A metal object at 120oF is set on an insulating pad to cool. The temperature falls from 120oF to 100oF in 12 minutes. The surroundings are 65oF. Find the time required for that object to continue to cool from 98oF to 80oF. Assume negligible conduction and radiation losses on both cases. 32. Assume that a body cools according to Newton’s Law of Cooling dT/dt = -kѲ, where t is the time and q is the difference between the temperature T of the body and that of the surrounding air. Find the temperature T at time t a boiler of water cools in air at 0oC if the water is initially boiling at 100oC and the temperature dropped 10oC during the first 20 minutes. Also, find the time for the temperature of water to drop from 90oC to 80oC and the temperature of water after 90 minutes. 33. Water at temperature 100oC cools in 10 minutes to 80oC in a room temperature of 25oC. a. Find the temperature of the water after 20 minutes and, b. When will the temperature be 40oC?

34. An object cools from 120oF to 95oF in half an hour when surrounded by air whose temperature is 80oF. Find the temperature at the end of another half an hour. 35. A pot of liquid is put on the stove to boil. The temperature of the liquid reaches 170oF and the pot is taken off the burner and placed on the counter in the kitchen. The temperature of the air in the kitchen is 76oF, after two minutes the temperature of the liquid in the pot is 123oF. How long before the temperature of the liquid in the pot will be 84oF? 36. A glass of hot milk at 100oC is brought in a room where the temperature is maintained at 20oC. After 15 minutes the temperature goes down to 90oC. Find: a. The temperature of the glass of milk after 15 more minutes. b. The time it takes until the temperature goes down to 37oC. 37. A body 300oK is brought to a room with air temperature equal to 350oK. After 1 minute in the room, the body is already 315oK. After how many minutes will the body become 340oK? 38. Water at temperature 10oC takes 5 minutes to warm up to 20oC in a room of 40oC temperature. a. Find the temperature after ½ hour. b. When will the temperature be 25oC? 39. A thermometer reading 75oF is taken out where the temperature is 20oF. The reading is 30oF 4 minutes later. Find a. the thermometer reading 7 minutes after it was brought outside, and b. the time taken for the reading to drop from 75oF within a half degree of the air temperature 40. If the temperature of the air is 290oK, a certain substance cools from 370oK to 330oK in 10 minutes. Find the temperature after 40 minutes. 41. The rate at which the substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 30oK and the substance cools from 370oK to 340oK in 15 minutes, when will the temperature be 310oK? 42. When the temperature reads 36oF, it is placed in the oven, After 1 and 2 minutes, it reads 60oF and 82oF. What is the temperature of the oven? 43. A tank containing 100 gallons of brine made by dissolving 80 lbs of salt in water. Pure water runs out into the tank at the rate of 4 gpm and the mixture kept uniform by stirring runs out at the same rate. Find the amount of salt in the tank at any time. 44. Brine from first tank runs into a second tank at 2 gallons per minute and brine from the second tank runs into the first tank at 1 gpm. Initially, there are 10 gallons of brine containing 20 lbs of salt in the

first tank and 10 gallons of freshwater in the second tank. How much salt will the first tank contain after 5 minutes? Assume that the brine in each tank is kept uniform by stirring. 45. A tank contains 50 gallons of water, brine containing 2 lbs per gallon of salt flows into the tank at the rate of 2 gpm. The mixture kept uniform by stirring runs out at the same rate. How long will it take before the quantity of the salt in the tank will be 50 lbs? 46. A tank has 60 gallons of pure water. A salt solution with 3 lbs of salt per gallon enters at 2 gal/min and leaves at 2.5 gal/min. a. Find the concentration of the salt in the tank at any time, b. find the salt concentration when the tank has 30 gallons of salt water, c. find the amount of water in the tank when the concentration is greatest and, d. determine the maximum amount of salt present at any time. 47. Chemical C is produced from a reaction involving chemical A and B. The rate of production of C varies as the product of the instantaneous amounts of A and B. The formation requires 3 lbs of A for every 2 lbs of B. If 60 lbs each of A and B are present initially and 50 lbs of C are formed in 1 hour find: a. the amount of C at any time, b. the amount of C after 2 hours and, c. the maximum quantity of C which can be formed. 48. Brine containing 2 lbs of salt per gallon runs into a tank at 2 gpm, brine solution from the first tank runs into the second tank at the rate of 3 gpm. Initially, the first tank contains 10 gallons of brine with 30 lbs of salt and 10 gallons of fresh water in the second tank. Assuming uniform concentration in each tank, find the quantity of salt in the second tank at the end of 5 minutes. 49. Brine from a first tank runs into a second tank at 5 gallons per minute, and brine from the second tank runs into the first tank at 4 gpm. Initially, there are 13 gallons of brine containing 23 lbs of salt in the first tank and 13 gallons of freshwater in the second tank. How much salt will the first tank contain after 10 minutes? Assume that the brine in each tank is kept uniform by stirring. 50. Chemical A is transformed into chemical B. The rate at which B is formed varies directly as the amount of A present at any instant. If 10 lbs of A is present initially and if 3 lbs is transformed into B in 1 hour, how much of A are formed after 2 hours? In what time is 75% of chemical A transformed? 51. A tank initially contains 200 liters of freshwater. Brine containing 2N/L dissolved salt enters the tank at 5L/min and the resulting mixture leaves the tank at the same rate. Find the salt concentration in the tank after 5 minutes. 52. A tank contains initially 2,500 liters of 50% salt solution; water enters the tank at the rate of 25 liters per minute. Find the percentage of salt in the tank after 30 minutes.