512 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions 50. Population Size Many environmental situations place effective limits on the growth of the number of an organism in an area. Many such limited-growth situations are described by the logistic function G1x2 = MG0 G0 + 1M- G02e-kMx , where G0 is the initial number present, M is the maximum possible size of the population, and k is a positive constant. The screens illustrate a typical logistic function calculation and graph. −5 −1 70 65 Assume that G0 = 100, M= 2500, k = 0.0004, and x = time in decades (10-yr periods). (a) Use a calculator to graph the function, using 0 … x … 8 and 0 … y … 2500. (b) Estimate the value of G122 from the graph. Then evaluate G122 algebraically to find the population after 20 yr. (c) Find the x-coordinate of the intersection of the curve with the horizontal line y = 1000 to estimate the number of decades required for the population to reach 1000. Then solve G1x2 = 1000 algebraically to obtain the exact value of x. Economics (Exercises 51–56) 51. Consumer Price Index The U.S. Consumer Price Index for the years 1981–2018 is approximated by A1t2 = 100e0.02698t, where t represents the number of years after 1981. (Since A1162 is about 154, the amount of goods that could be purchased for $100 in 1981 cost about $154 in 1997.) Use the function to determine the year in which costs will be 220% higher than in 1981. (Data from U.S. Bureau of Labor Statistics.) 52. Product Sales Sales of a product, under relatively stable market conditions but in the absence of promotional activities such as advertising, tend to decline at a constant yearly rate. This rate of sales decline varies considerably from product to product, but it seems to remain the same for any particular product. The sales decline can be expressed by the function S1t2 = S0 e-at, where S1t2 is the rate of sales at time t measured in years, S0 is the rate of sales at time t = 0, and a is the sales decay constant. (a) Suppose the sales decay constant for a particular product is a = 0.10. Let S0 = 50,000 and find S112 and S132 to the nearest thousand. (b) Find S122 and S1102 to the nearest thousand if S0 = 80,000 and a = 0.05. 53. Product Sales Use the sales decline function given in Exercise 52. If a = 0.10, S0 = 50,000, and t is time measured in years, find the number of years it will take for sales to fall to half the initial sales. Round the answer to the nearest tenth. 54. Cost of Bread Assume the cost of a loaf of bread is $4. With continuous compounding, find the number of years, to the nearest tenth, it would take for the cost to triple at an annual inflation rate of 4%.
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