REVIEW: LESSONS R-18 WORD PROBLEMS FALL 2018

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1 REVIEW: LESSONS R-18 WORD PROBLEMS FALL 2018 Lesson R: Review of Basic Integration 1. The growth rate of the population of a county is P (t) = t(4085t ), where t is times in years. How much does the population increase from t = 1 year to t = 4 years? Answer: 91, A faucet is turned on at 9:00am and water starts to flow into a tank at a rate of r(t) = 9 t, where t is time in hours after 9:00am and the rate r(t) is in cubic feet per hour. (a) How much water, in cubic feet, flows into the tank from 10:00am to 1:00pm? Answer: 42 cubic feet (b) How many hours after 9:00am will there be 187 cubic feet of water in the tank? Round your answer to the nearest tenth. Answer: 9.9 hours Lesson 1: Integration by Substitution (I) 3. It is estimated that t-weeks into the semester, the average amount of sleep a college math student gets per day S(t) decreases at a rate of 3t hours per day. When the semester begins, math students sleep an average of 8.7 hours per day. What is S(t), 14 weeks into the semester? (Express your answer to three decimal places.) Answer: hours per day 4. A pork roast is removed from the freezer and left on the counter to defrost. The temperature of the pork roast is 4 C when it was removed from the freezer, and t hours later was increasing at a rate of T (t) = 10.6e 0.3t C/hour. Assume the pork is defrosted when its temperature reaches 11 C. How long does it take for the pork roast to defrost? (Estimate answer rounded off to 4 decimal places.) 1 e t2

2 2 REVIEW: LESSONS R-18 WORD PROBLEMS Answer: hours 5. The velocity v(t) of an ant that is moving along the t-axis is given by: 3t v(t) = (19 + t 2 ). 3/2 The position s(t) of the ant at time t = 0 is 23. What is s(t) at time t? Answer: s(t) = 3 t It is estimated that t years from now, the value of a small piece of land, V (t), will be increasing at a rate of 2t 3 0.4t dollars per year. The land is currently worth $ 560. Find the value of the land after 10 years to the nearest cent. Answer: V (10) = dollars Lesson 2: Integration by Substitution (II) 7. After t months on the job, a postal clerk can sort Q(t) = e 0.5t letters per hour. What is the average rate at which the clerk sorts mail during the first 3 months on the job? Round your answer to two decimal places. Answer: letters per hour 8. If the area of the region under the curve y = 1 7x + 3 over the interval 0 x a is 10, then what is a? Round your answer to 3 decimal places. Answer: A certain plant grows at a rate H (t) = 1 3 9t + 2 inches per day, t days after it was planted. How many inches will the height of the plant change on the third day? Round your answer to the nearest thousandth. Answer:.345 inches

3 REVIEW: LESSONS R-18 WORD PROBLEMS It is estimated that t hours after 8:00am, the population of a certain bacterial sample will be changing at a rate of N (t) = 3t t + 3 hundred bacteria per hour. Find the increase in the bacteria population from 11:00am to 2:00pm. Round your answer to three decimal places. Answer: hundred bacteria 11. Suppose that as a yellow car brakes, its velocity is described by v(t) = 2.3e 1 t.07 meters/second. If the brakes are applied at time t = 0 seconds, what is the distance it takes for the car to come to a complete stop. Round your answer to 3 decimal places. Answer: meters 12. A science geek brews tea at 195 F, and observes that the temperature T (t) of the tea after t minutes is changing at the rate of T (t) = 3.5e 0.04 F /min. What is the average temperature of the tea during the first 16 minutes after being brewed? Round your answer to the nearest hundredth of a degree. Answer: F Lesson 4: The Natural Logarithmic Function Integration 13. A factory is discharging pollution into a river at the rate p = 9t 2t tons per year, where t is the number of years after the factory starts operations. Find the total amount of pollution discharged during the first 13 years of the factory s operations. Round your answer to 3 decimal places. Answer: 8.77 tons 14. The selling price for a product is p = 118, x where p is the price (in dollars) and x is the number of units (in thousands). Find the average price on the interval 70 x 80. Round your answer to 2 decimal places. Answer: dollars 15. It is estimated that t weeks from now, the average price of a gallon of milk will be increasing at the rate of p (t) = t t

4 4 REVIEW: LESSONS R-18 WORD PROBLEMS If the average price of a gallon of milk is currently $ 1.85, what will the average price of a gallon of milk be 6 weeks from now? Round your answer to the nearest thousandth of a dollar. Answer: dollars 16. A beautiful ice sculpture is melting, with its volume changing at a rate of V (t) = 60 t( t 10) ft 3 /hr, for 0 < t < 90. If after 1 hour, the volume of the sculpture is 260 ft 3, find its volume after 64 hours. If the sculpture has melted completely, put 0. Round your answer to 3 decimal places. Answer: ft The rate of sales for a corporation t weeks from now is given by S(t) = 5 5t + 4 millions of dollars per week. Find the average sales per week for the first 6 weeks. Round your answer to the nearest dollar. Answer: 356, 678 dollars Lesson 5: Integration by Parts (II) 18. A certain chemical reaction produces a compound X at a rate of 10t (t + 4) 3 kg/hour, where t is time (in hours) from the start of the reaction. How much of the compound is produced during the first five hours of the reaction? Round your answer to the nearest hundredth of a kilogram. Answer: kg 19. A model for the ability of a child to memorize information, measured on a scale from 1 to 100, is given by M = 1 + 4t ln(t); 2 t 8 where t is the child s age in years. Find the child s average memorization ability between ages 2 and 5. Round your answer to three decimal places. Answer: The velocity of a car over the time period 0 t 3 is given by the function v(t) = 45e t/4 miles per hour, where t is time in hours. What was the distance the car traveled in the first 30 minutes? Round your answer to two decimal places. Answer: 5.18 miles

5 REVIEW: LESSONS R-18 WORD PROBLEMS A certain plant grows at the rate of 4 ln( t + 1) (t + 1) 2 feet per year, t years after it is planted. If the plant was 3.9 feet tall when it was planted, then how tall will it be in 2 years? Round your answer to the nearest thousandth. Answer: feet 22. The number of members of a newly founded club is M(t) = 100(5te2t + 3) e t members, where t is the number of months after its opening in the beginning of 2010, and t = 0 corresponds to January 1. What is the average number of members during the period comprised of February and March that year? (Assume that all months have equal duration). Round your answer to the nearest integer. Answer: 10, 090 members 23. When samples of iron ore are tested for potential mining sites, the probability (0 to 1) of finding sample that has x percentage of iron in the sample is described by ( ) x, 1 + 7x where x is also between 0 and 1. Find the probability that a tested sample of iron ore is at least 52% iron. Round your answer to 4 decimal places. Answer:.3454 Lesson 6: Differential Equations 24. Let y(t) denote the mass of a radioactive substance at time t. Suppose this substance obeys the equation y (t) = 14y(t). Assume that, initially, the mass of the substance is y(0) = M > 0. At what time does half of the mass remain? (Round your answer to the 3 decimal places). Answer: t = The rate of change of the number of coyotes N(t) is directly proportional to 650 N(t), where t is the time in years. That is, dn dt = k(650 N). When t = 0, the population is 290, and when t = 2, the population has increased to 520. Find the population when t = 3. (Round your answer to the nearest whole number.) Answer: 572

6 6 REVIEW: LESSONS R-18 WORD PROBLEMS 26. A bacterial culture grows at a rate proportional to its population. If the population is 4000 at t = 0 and 6000 at t = 1 hours, find the population as a function of time. Answer: y = 4000e ln(3/2)t 27. A radioactive element decays with a half-life of 6 years. If the mass of the element weighs 7 pounds at t = 0, find the amount of the element after 13.9 years. Round your answer to 4 decimal places. Answer: Write a differential equation describing the following situation: the rate at which people become involved in a corporate bribing scheme is jointly proportional to the number of people already involved and the number of people who are not yet involved. Suppose there are a total of 1900 people in the company. Use k for the constant, P for the number of people who are involved in the scheme, and t for time. Answer: dp dt = k(1900 P )P 29. After 10 minutes in Jean-Luc s room, his tea has cooled to 45 Celsius from 100 Celsius. The room temperature is 25 Celsius. How much longer will it take to cool to 37? (Round your answer to the nearest hundredth). Answer: 3.86 minutes Lesson 7: Differential Equations Separation of Variables (I) 30. When an object is removed from a furnace and placed in an environment with a constant temperature of 80 F, its temperature is 1, 651 F. One hour after it is removed, the temperature of the object is 1, 069 F. Find the temperature of the object 3 hours after the object is removed from the furnace. Round your answer to 2 decimal places. Answer: T = F 31. A teen chewing bubble gum blows a huge bubble, the volume of which satisfies the differential equation dv dt = 3 3 V 2. Where t is the time in seconds after the teen start to blow the bubble. If the bubble pops as soon as it reaches 512 cubic centimeters in volume, how many seconds does it take for the bubble to pop? Assume that the bubble had no volume when the teen first started blowing. Answer: 8 seconds

7 REVIEW: LESSONS R-18 WORD PROBLEMS A wet towel hung on a clothesline to dry outside loses moisture at a rate proportional to its moisture content. After 1 hour, the towel has lost 30% of its original moisture content. After how long will the towel have lost 75% of its moisture content? Round your answer to 3 decimal places. Answer: t = hours Lesson 8: Differential Equations Separation of Variables (II) 33. A 600-gallon tank initially contains 300 gallons of brine containing 75 pounds of dissolved salt. Brine containing 3 pounds of salt per gallon flows into the tank at a rate of 4 gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 1 gallon per minute. Set up a differential equation for the amount of salt, A(t), in the tank at time t. Answer: da dt = 12 A 3t In a particular chemical reaction, a substance is converted into a second substance at a rate proportional to the square of the amount of the first substance present at any time, t. Initially, 43 grams of the first substance was present, and 1 hour later only 14 grams of the first substance remained. What is the amount of the first substance remaining after 6 hours? (Round your answer to 4 decimal places). Answer: grams 35. The rate of change in the number of miles of roads cleared per hour by a snowplow with respect to the depth of the snow is inversely proportional to the depth of the snow. Given that 23 miles per hour are cleared when the depth of the snow is 2.1 inches and 13 miles per hour are cleared when the depth of the snow is 8 inches, then how many miles of the road will be cleared each hour when the depth of the snow is 11 inches? Round your answers to 3 decimal places. Answer: miles each hour 36. A 500-gallon tank initially contains 300 gallons of pure distilled water. Brine containing 4 pounds of salt per gallons flows into the tank at the rate of 5 gallons per minute, and the well stirred mixture flows out of the tank at the rate of 5 gallons per minute. Find the amount of salt in the tank after 10 minutes. Round your answer to three decimal places. Answer: pounds Lesson 10: First-Order Linear Differential Equations (II) 37. Pam owns an electronic store with storage capacity for 90 computer tablets. She currently has 65 computer tablets in inventory and determines that they are selling at a daily rate equal to 13% of the available capacity. When will Pam sell out of computer tablets? (Round your answer to the 3 decimal places).

8 8 REVIEW: LESSONS R-18 WORD PROBLEMS Answer: days 38. In Purdue s Chemistry department, the chemists have found that in a water based solution containing 10 grams of a certain undissolved chemicals, the rate of change of the amount of chemicals dissolved in the solution is proportional to the amount of the undissolved chemicals. Let Q(t) (in grams) be the amount of dissolved chemicals at time t and let k be the positive proportionality constant. Find the differential equation describing this given situation. Answer: dq dt = k(10 Q) 39. A corporation is initially worth 6 million dollars and is growing in value, V, by 22% each year, and is additionally gaining 24% of a growing market estimated at 100e 0.22t million dollars, where t is the number of years the company has existed. Approximate the value of the company after 7 years. (Round your answer to the nearest million dollars.) Answer: 812 million 40. A 500-gallon tank initially contains 200 gallons of brine containing 65 pounds of dissolved salt. Brine containing 3 pounds of salt per gallon flows into the tank at the rate of 4 gallons per minute, and the well-stirred mixture flows out of the tank at a rate of 1 gallon per minute. Set up a differential equation for the amount of salt A(t) in the tank at time t. How much salt is in the tank when it is full? Round your answer to the 2 decimal places. Answer: 1, pounds 41. A corporation starts to invest part of its revenue continuously at rate of P dollars per year in a fund for future expansion plans. Assume the fund earns money at an annual interest rate, r, compounded continuously. The rate of growth of the amount A in the fund given by da dt = ra + P. Find A as a function of time of P = 200 and r = 17%. Round coefficients and constants to two decimal places as needed. Answer: A = 1, e 0.17t 1, A 21,000 cubic foot room initially has a radon level of 870 picocuries per cubic foot. A ventilation system is installed that brings in 700 cubic feet of air per hour that contains 7 picocuries per cubic foot, while an equal quantity of wellmixed air in the room leaves the room each hour. How long will it take for the room to reach a safe to breathe level of 120 picocuries per cubic foot. Round your answer to 5 decimal places. Answer: hours

9 REVIEW: LESSONS R-18 WORD PROBLEMS 9 Lesson 11: Area of a Region between Two Curves 43. Given two revenue models (in millions of dollars) for a large corporation, where both models are estimates of revenues from 2015 through 2040 and t = 0 corresponds to the beginning of 2015: R 1 = t t 2 R 2 = t t 2 which model predicts the greater revenue? How much more revenue does the greater revenue model predict over the 5 year period from 2025 through 2029? Round your answer to 3 decimal places. Answer: millions of dollars 44. A company reports that profits for the past fiscal year were 14.2 million dollars. Given that t is the number of years from now, the company predicts that profits will grow continuously for the next 7 years at a continuous annual rate between 3.6% and 5.8%. Estimate the positive cumulative difference in predicated total profits over the 7 years based on the predicted range of growth rates. Round your answer to 3 decimal places. Answer: million dollars 45. A factor installs new machinery that will save x dollars per year, where x is the number of years since installation. However, the cost of maintaining the new machinery is 91x dollars per year. Find the accumulated savings that will occur before the machinery should be replaced. Round your answer to 3 decimal places. Answer: 5, dollars 46. Find the equation of the vertical line that divides the area of the region R bounded by y = 1 10x 2, y = x, and y = x 16 Round your answer to 4 decimal places. Answer: x =.456 in half. Lesson 12: Volume of Solids of Revolution (I) 47. The shape of a fuel tank for the wing of a jet aircraft is designed by revolving the region bounded by the function y = 10 7 x2 5 x and the x-axis, where 0 x 5, about the x-axis. Given x and y are in meters, find the volume of the fuel tank. Round your answer to 2 decimal places. Answer: 3, m 3

10 10 REVIEW: LESSONS R-18 WORD PROBLEMS Lesson 15: Improper Integrals 48. Suppose a nuclear accident causes plutonium to be released into the atmosphere. The total amount of energy that has been released by time a is given by a 0 7e 3t dt. What is the total amount of energy that will be given off over all time? Round your answer to 3 decimal places. Answer: Lesson 17: Geometric Series and Convergence (II) 49. Suppose that in a country, 40% of all income the people receive is spent, and 60% is saved. If saving habits do not change and continue indefinitely, what is the total amount of spending generated in the long run by a 36 billion dollar tax rebate which is given to the country s citizens to stimulate the economy? Note: Include the entire government tax rebate as part of the total spending. (a) Enter the four largest terms of the infinite series in order from larger to smaller as numbers. Round your answer to 3 decimal places. Answer: billions (b) What is the total amount of spending generated? Round your answer to two decimal places. Answer: 60 billions 50. A ball has the property that each time it falls from a height h onto the ground, it will rebound to a height of rh, where r (0 < r < 1) is called the coefficient of restitution. Find the total distance traveled by a ball with r = 0.4 that is dropped from a height of 17 meters. If the ball rebounds indefinitely, approximately how many meters will the ball travel? (a) Enter the 6 largest terms of the infinite series in order from larger to smaller as numbers. Round your answers to 4 decimal places. Answer: meters (b) What is the total distance the ball traveled? Round your answer to 3 decimal places. Answer: meters 51. In a right triangle, a series of altitudes are drawn starting with an altitude drawn using the vertex of the right angle and drawn towards the hypotenuse.

11 REVIEW: LESSONS R-18 WORD PROBLEMS 11 Then subsequently continuing to draw altitudes from the right angles in the new right triangles created, and which also include the vertex from the smallest angle in the original right triangle. The series of altitudes are drawn so they move closer and closer to the smallest angle in the original right triangle. Find the sum of the lengths of these altitudes given that one of the angles of the original triangle is 55 degrees and the side of the triangle adjacent to this angle is 16.8 meters long. If the process continues indefinitely, what will be the total sum of the altitudes? (a) Enter the largest terms of the infinite series in order from larger to smaller as numbers. Round your answers to 4 decimal places. Answer: meters (b) What is the total sum of the altitudes? Round your answer to 3 decimal places. Answer: meters 52. A patient is given an injection of 50 milligrams of a drug every 24 hours. After t days, the fraction of the drug remaining in the patient s body is f(t) = 2 t/5. If the treatment is continued indefinitely, approximately how many milligrams of the drug will eventually be in the patient s body just prior to an injection? (a) Enter the three largest terms of an infinite series in order from largest to smallest as numbers. Round your answer to 4 decimal places. Answer: milligrams (b) Approximately how many milligrams of the drug will eventually be in the patient s body just prior to an injection? Round your answer to 3 decimal places. Answer: milligrams 53. How much money (in dollars) should you invest today at an annual interest rate of 5.5% compounded continuously so that, starting two years from today, you can make annual withdrawals of $ 3,500 in perpetuity? (a) Enter the three largest terms of the infinite series in order from larger to smaller as numbers. Round your answer to 3 decimal places. Answer: dollars (b) What is the total dollar amount you should invest today? Round your answer to the nearest cent. Answer: 58, dollars

12 12 REVIEW: LESSONS R-18 WORD PROBLEMS people are sent to a colony on Mars, and each subsequent year 760 more people are added to the population of the colony. The annual death proportion is 8%. Suppose this pattern continues indefinitely. (a) Enter the largest terms in the infinite series in order from larger to smaller as numbers that describes the situation. Round your answers to 3 decimal places. Answer: people (b) What is the eventual population on Mars? Round your answer to the nearest whole number. Answer: 8740 people Lesson 18: Functions of Several Variables 55. A chain of paint store carries two brands of latex paint. Sales figure indicate that if the first brand is sold for x dollars per gallon and the second for y dollars per gallon, the demand for the first brand will be D 1 (x, y) = x + 30y gallons per month and the demand for the second brand will be D 2 (x, y) = x 20y gallons per month. (a) Express the chain of paint store s monthly revenue from the sale of paint as a function of x and y. Answer: R(x, y) = x(30y 20x + 600) + y( 20y + 80x + 400) (b) Compute the revenue from part (a) if the first brand is sold for $ 21 per gallon and the second for $ 15 per gallon. Answer: $39, 930