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Margaret L. Lial,Thomas W. Hungerford, John P. Holcomb. Mathematics with Applications: In the Management, Natural, and Social Sciences, 10th Edition. Boston: Addison Wesley, 2011.

Section 4.1

35. Finance. If $1 is deposited into an account paying 6% per year compounded annually, then after t years the account will contain y = (1 + .06)t = (1.06)t dollars. (a) Use a calculator to complete the following table. (b) Graph y = (1.06)t

36. Finance. If money loses value at the rate 3% per year, the value of $1 in t years is given by y = (1 - .03)t = (0.97)t dollars. (a) Use a calculator to complete the following table. (b) Graph y = (0.97)t
37. Finance. If money loses value, then as time passes, it takes more dollars to buy the same item. Use the results of Exercise 36(a) to answer the following questions. (a) Suppose a house costs $105,000 today. Estimate the cost of the same house in 10 years. (b) Estimate the cost of a $50 textbooks in 8 years.

38. Natural Science. Biologists have found that the oxygen consumption of yearling salmon is given by g(x) = 100e0.7x, where x is the speed in feet per second. Find each of the following. (a) the oxygen consumption when the fish are still; (b) the oxygen consumption when the fish are swimming at a speed of 2 feet per second.

39. Business. Cell phone usage has been increasing exponentially in recent years. The number of U.S. cell phone accounts (in millions) is approximated by f(x) = 110.6e0.125x (0 ≤ x ≤ 7), where x = 0 corresponds to the year 2000. Estimate the number of cell phone accounts in (a) 2002 (b) 2004 (c) 2006 (d) 2010 (e) Explain why your answer to part (d) is not likely to be accurate.

40. Business. The monthly payment on a car loan at 12% interest per year on the unpaid balance is given by f(n) = P/((1 – 1.01-n)/0.01) where P is the amount borrowed and n is the number of months over which the loan is paid back. Find the monthly payment for each of the following loans. (a) $8000 for 48 months (b) $8000 for 24 months (c) $6500 for 36 months (d) $6500 for 60 months.

41. Natural Science. The amount of plutonium remaining from 1 kg after x years is given by the function W(x) = 2-x/24360. How much will be left after (a) 1000 years (b) 10000 years (c) 15000 years
(d) Estimate how long it will take for the 1 kilogram to decay to half its original weight.

Business. The scrap value of a machine is the value of the machine at the end of its useful life. By one method of calculating scrap value, where it is assumed that a constant percentage of value is lost annually, the scrap value is given by S = C(1 – r)n, where C is the original cost, n is the useful life of the machine in years, and r is the constant annual percentage of value lost. Find the scrap value for each of the following machines.
42. Original cost, $68,000; life, 12 years; annual rate of value loss, 15%.

45. There were fewer than a billion people on Earth when Thomas Jefferson died in 1826, and there are now more than 6 billion. If the world population continues to grow as expected, the population (in billions) in year t will be given by the function P(t) = 4.834(1.01(t – 1980)). Estimate the world population in the following years. (a) 2005 (b) 2010 (c) 2030 (d) What will the world population be when you reach 65 years old.

48. Business. Construction of expressways in China has expanded significantly in recent decades. The approximate number of kilometers of expressways in China (in thousands) is given by f(x) = 1.2e0.228x (0 ≤ x ≤ 20), where x = 0 corresponds to the year 1990. How many kilometers of expressways did China have in the given years (a) 2006 (b) 2007 (c) Assuming that this function remains valid, use technology to determine when there are 500000 kilometers of expressways in China.

49. Health. The United States has been experiencing a shortage of registered nurses, and the situation is expected to worsen in coming years. The projected shortage of fulltime equivalent nurses (in thousands) can be approximated by N(x) = 100.5(1.11)x where x = 0 corresponds to the year 2000. Find the shortage in the given years. (a) 2010 (b) 2013 (c) 2015 (d) Use technology to find the year in which the shortage will exceed 6000,000 nurses.

50. Business. The amount spent per person per year on cable and satellite TV can be approximated by g(x) = 497/(1 + 1.884e-0.183x), where x = 0 corresponds to the year 2000. Find the amount per person spent in the given years. (2005) (b) 2010 (c) When will per-person spending reach $425?

51. Business. Sales of music CDs (in million of dollars) in the United States are approximated by f(t) 13.82(0.935)t, where t = 0 corresponds to the year 2000. How much is spent on CDs in the given years? (a) 2002 (b) 2006 (c) Graph f(t) for the period from 2000 to 2015.

Section 4.2

1. Finance. Suppose you owe $800 on your credit card and you decide to make no new purchases and to make the minimum monthly payment on the account. Assuming that the interest rate on your card is 1% per month on the unpaid balance and that the minimum payment is 2% of the total (balance plus interest), your balance after t months is given by B(t) = 800(0.9898t). Find your balance at each of the given times. (a) six months (b) one year (c) five years (d) eight years (e) On the basis of your answers to parts (a)-(d), what advice would you give to your friends about minimum payments?

2. Health. The amount spent per person on health care in the United States is approximated by H(t) = h01.065t, where t = 0 corresponds to the year 2000 and H(t) is in thousands of dollars. (a) The amount spent per person in 2000 was $4624. Find h0. (b) To the nearest dollar, estimate the per-person costs in 2008, 2010, and 2012.

15. Boiling water at 100 °Celsius is placed in a freezer at -18 °Celsius. The temperature of the water is 50 °Celsius after 24 minutes. Find the temperature of the water after 76 minutes.

16. Paisley refuses to drink coffee cooler than 95 °F. She makes coffee with a temperature of 170 °F in a room with a temperature of 70 °F. The coffee cools to 120 °F in 10 minutes. What is the longest amount of time she can let the coffee sit before she drinks it?

17. Social Sciences. A sociologist has shown that the fraction y(t) of people in a group who have heard a rumor after t days is approximated by y(t) = y0ekt/(1 – y0(1 - ekt)), where y0 is the fraction of people who heard the rumor at time t = 0 and k is a constant. A graph of y(t) for a particular value of k is shown in the figure. (a) If k = .1 and y0 = .05, find y(10). (a) If k = .2 and y0 = .10, find y(5). (c) Assume the situation in part (b). How many weeks will it take for 65% of the people to have heard the rumor?

Section 4.3

62. Two people with flu visited a college campus. The number of days, T, that it took for the flu virus to infect n people is given by the equation T= -1.43 ln ((10,000 - n) / 4998n). How many days will it take for the virus to infect (a) 500 people? (b) 5000 people?

63. Business. Domestic sales of DVDs and videotapes (in billions of dollars) are approximated by g(x) = 10.155 + 3.62 ln x, where x = 1 corresponds to the year 2001. (a) Estimate the sales in 2004 and 2008. (b) Graph the function g for the period from 2001 to 2025. (c) Assuming that this model remains accurate, what does the shape of the graph suggest about DVD and videotape sales?

Chapter 6

1. The U-drive Company in Example 1 learns that each van now costs $13000 and each small track $18000. The company decides to buy only 182 new vehicles. How many of each kind should it buy?

2. Suppose that Ellen McGillicuddy in Example 2 finds that her annual return on the international stock fund will be only 4%. Now how much should she put in each investment?

3. To meet consumer demand, the animal feed in Example 3 now supplies only 2400 units of fat. How many units of corn, soybeans, and cottonseed are now needed?

4. Suppose that Kelly Karpet Kleaners in Example 5 finds a way to pack the EZ model in an 8-cubic-foot box. Now how many of each model should a fully loaded van carry?

5. Matt took clothes to the cleaners three times last month. First, he brought 3 shirts and 1 pair of slacks and paid $10.96. Then he brought 7 shirts, 2 pairs of slacks, and a sports coat and paid $30.40. Finally, he brought 4 shirts and 1 spots coat and paid $14.45. How much was he charged for each shirt, each pair of slacks, and each sports coat?

6. Business. A minor league baseball park has 7000 seats. Box seats cost$6, grandstand seats cost $4, and bleacher seats cost $2. When all seats are sold, the revenue is $26,400. If the number of box seats is one-third the number of bleacher seats, how many seats of each type are there?

7. Business. Tickets to a band concert cost $5 for adults, $3 for teenagers, and $2 for preteens. There were 570 people at the concert, and total ticket receipts were $1950. Three-fourth as many teenagers as preteens attended. How many adults, teenagers, and preteens attended?

8. Shipping charges at an online bookstore at $4 for 1 book, $6 for 2 books and $7 for three to five books. Last week there were 6400 orders of five or fewer books, and total shipping charges for these orders were $33,600. The number of shipments with $7 charges was 1000 less than the number with $6 charges. How many shipments were made in each category?

10. Finance. Kate borrows $10000, some from her friends at 8% annual interest, twice as much as that from her bank at 9%, and the remainder from her insurance company at 5%. She pays a total of $830 in interest for the first year. How much did she borrow form each source?

11. Business. Pretzels cost$3 per lb, dried fruit $4 per pound, and nuts $8 per pound. How many pounds of each should be used to produce 140 pounds of trail mix costing $6 per pound in which there are twice as many pretzels (by weight) as dried fruit?

24. Business. An electronics company produces transistors, resistors and computer chips. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Each resistor requires 3, 2 and 1 units of the three materials, and each computer chip requires 2, 1, and 2 units of these materials, respectively. How many of each product can be made with the following amount of materials? (a) 810 units of copper, 410 of zinc, and 490 of glass (b) 765 units of copper, 385 of zinc, and 470 of glass (c) 1010 units of copper, 500 of zinc, and 610 of glass.

32. Management. There are three convenience stores in Gambier. This week, Store I sold 88 loaves of bread, 48 quarts of milk, 16 jars of peanut butter and 112 pounds of cold cuts. Store II sold 105 loaves of bread, 72 quarts of milk, 21 jars of peanut butter, and 147 pounds of cold cuts. Store III sold 60 loaves of bread, 40 quarts of milk, no peanut butter, and 50 pounds of cold cuts. (a) Use a 3 4 matrix to express the sales information for the three stores. (b) During the following week, sales on these products increased by 25%; sales at Store II increase by One-third, and sales at Store III increased by 10%. Write the sales matrix for that week. (c) Write a matrix that represents total sales over the two-week period.

54. Business. Burger Barn's three locations sell hamburgers, fries, and soft drinks. Barn I sells 900 burgers, 600 orders of fries, and 750 soft drinks each day. Barn II sells 1500 burgers a day and Barn III sells 1150. Soft drink sales number 900 a day at Barn II and 825 a day at Barn III. Barn II sells 950 orders of fries per day and Bam III sells 800. (a)Write a 3 3 matrix S that displays daily sales figures for all locations. (b)Burgers cost $1.50 each, fries $.90 an order, and soft drinks $.60 each. Write a 1 3 matrix P that displays the prices. (c) What matrix product displays the daily revenue at each of the three locations? (d)What is the total daily revenue from all locations?

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