SECTION 5.3 Equations and Applications with Exponential and Logarithmic Functions 349 | EXERCISES | 5.3 In Problems 1-22, solve each equation. Give answers correct to 3 decimal places in Problems 1-12. 1. 834 = 32,768 2. 72% = 823,543 3. 0.13P = P(2-*) 4. 0.05A = A(1.06) - 5. 25,000 =10,000(1.05)2X 6. 75,000-15,000(1.02) 7. 10,000 = 1500e 10x 8. 2500 = 600005 9. 78 = 100 - 1000 DOLX 10. 500 = 600 - 600e -0.4% 60 200 11. 55 = 12. 150 = 1 + 5e-1.6% 1 + 30e-9.30 13. log x = 5 14. In x=8 15. log, (9x + 1) = 3 16. log(x + 2) = 2 17. In x - In 5 = 10 18. In x + In 8 = 5 19. 7 + log(8x) = 25 - 2 log x 20. 3 log x + 10 = log(3x) + 14 21. In (x + 2) + In x = ln (x + 12) 22. log, * - log, (x + 3) = log, (x - 2) (a) How long will it be before a pension of $60,000 per year has a purchasing power of $30,000? (b) How much pension A would be needed so that the purchasing power Pis $50,000 after 15 years? 28. Product reliability A statistical study shows that the fraction of television sets of a certain brand that are still in service after x years is given by f(x) = 0.15* (a) What fraction of the sets are still in service after 5 years? (b) After how many years will the fraction still in service be 1/10? 29. Radioactive half-life An initial amount of 100 g of the radioactive isotope thorium-234 decays according to Q(t) = 100e -0.02828 where t is in years. How long before half of the initial amount has disintegrated? This time is called the half-life of this isotope. 30. Radioactive half-life A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t) = A e where A(t) is the amount of the isotope at time t, in years, and A, is the original amount. (a) If A, = 500 lb, how much will be left after a human lifetime (use t = 80 years)? (b) Find the half-life of this isotope. 31. Population growth If the population of a certain county was 100,000 in 1998 and 110,517 in 2008, and if the formula y = P, et applies to the growth of the county's population, estimate the population of the county in 2023. 32. Population growth The population of a certain city grows according to the formula y = P, 2003. If the population was 250,000 in 2000, estimate the year in which the population reaches 350,000. 33. Health care for the years from 2006 and projected to 2021, the national health care expenditures H, in billions of dollars, can be modeled by А For Problems 23 and 24, let f(x) = Use a 1 + ce-*' graphing utility to make the requested graphs. 100 23. (a) Fix A = 100 and graph y = f(x) = 1 + ce * for c = 0.25, 1, 9, and 49. (b) What effect does chave on the graphs? A 24. (a) Fix c = 1 and graph y = f(x) = for A = 50, 100, and 150. (b) What effect does A have on the graphs? -0.000028761 1 +e APPLICATIONS 25. Sales decay The sales decay for a product is given by S = 50,000 OM, where is the monthly sales and x is the number of months that have passed since the end of a promotional campaign. (a) What will be the sales 4 months after the end of the campaign? (b) How many months after the end of the campaign will sales drop below 1000, if no new campaign is initiated? 26. Sales decay The sales of a product decline after the end of an advertising campaign, with the sales decay given by S = 100,000e 0.5*, where S represents the weekly sales and x represents the number of weeks since the end of the campaign. (a) What will be the sales for the tenth week after the end of the campaign? (b) During what week after the end of the campaign will sales drop below 400? 27. Inflation The purchasing power P (in dollars) of an annual amount of A dollars after t years of 5% infla- tion decays according to H = 20090051941 where t is the number of years past 2005 (Source: U.S. Department of Health and Human Services). When are national health care expenditures expected to reach $4.0 trillion (that is, $4000 billion)? 34. U.S. debt For selected years from 1900 to 2013, the national debt d, in billions of dollars, can be modeled by d = 1.600.0834 P = Ae-0.154 where t is the number of years past 1900 (Source: Bureau of Public Debt, U.S. Treasury). How long will it be before the debt is predicted to reach $30 trillion (that is, $30,000 billion)? (Source: Viewpoints, VALIC) 350 CHAPTER 5 Exponential and Logarithmic Functions (a) What is the amount after 18 months? (b) How long before the investment doubles? 44. Compound interest If $1000 is invested at 10% com- pounded continuously, the future value S at any timet (in years) is given by S = 10000 (a) What is the amount after 1 year? (b) How long before the investment doubles? 45. Compound interest If $5000 is invested at 9% per year compounded monthly, the future value S at any timet (in months) is given by S= 5000(1.0075)'. (a) What is the amount after 1 year? (b) How long before the investment doubles? 46. Compound interest If $10,000 is invested at 1% per month, the future value S at any time t (in months) is given by S = 10,000(1.01). (a) What is the amount after 1 year? (b) How long before the investment doubles? Profits An investment services company experienced dramatic growth in the last two decades. The following models for the company's revenue R and expenses or costs (both in millions of dollars) are functions of the years past 1990. R(t) = 21.40.131: and C(t) = 18.6e0.1311 35. Demand The demand function for a certain com- modity is given by p = 100e-42. (a) At what price per unit will the quantity demanded equal 6 units? (b) If the price is $1.83 per unit, how many units will be demanded, to the nearest unit? 36. Demand The demand function for a product is given by p= 3000e-4/3. (a) At what price per unit will the quantity demanded equal 6 units? (b) If the price is $149.40 per unit, how many units will be demanded, to the nearest unit? 37. Supply If the supply function for a product is given by p = 100e" /(q + 1), where q represents the number of hundreds of units, what will be the price when the producers are willing to supply 300 units? 38. Supply If the supply function for a product is given by P = 200(24), where q represents the number of hundreds of units, what will be the price when the producers are willing to supply 500 units? 39. Total cost The total cost function for x units of a product is given by C(x) = 2500 ln (2x + 1) + 1500 (a) Find the total cost of producing 80 items. (b) How many units can be produced before total costs reach $16,000? 40. Total cost The total cost function for a product is C(x) = 800 In (x + 10) + 1700 where x is the number of units produced. (a) Find the total cost of producing 100 units. (b) Producing how many units will give total costs of $7500? 41. Demographics The millions of White non-Hispanic individuals in the U.S. civilian non-institutional popu- lation 16 years and older for selected years from 1980 and projected to 2050 can be modeled by the function y = 96.12 + 17.43 In x where x is equal to the number of years past 1970 (Source: U.S. Census Bureau). Find the year in which the number of White non-Hispanics is expected to reach 166.5 million. 42. Diabetes Centers for Disease Control and Prevention data from 2010 and projected to 2050 indicate that adult diabetes in the United States could dramatically increase. With x = 0) in 2000, the percent of U.S. adults with diabetes can be modeled by y = -13.0 + 11.9 In x Use the model to find the year in which the percent of U.S. adults with diabetes is predicted to reach 30%. 43. Compound interest If $8500 is invested at 11.5% com- pounded continuously, the future value S at any timet (in years) is given by S = 8500e1.115 Use these models in Problems 47 and 48. 47. (a) Use the models to predict the company's profit in 2020. (b) How long before the profit found in part (a) is predicted to double? 48. Use the models to find how long before the company's profit reaches $500 million. 49. Purchasing power Using Social Security Administration data for selected years from 2012 and projected to 2050, with a purchasing power of $1.00 in 2012, the function P(x) = 1.078(1.028) gives the purchasing power of a 2012 dollar as a func- tion of x, the number of years past 2010. (a) Find and interpret P(18). (b) Solve algebraically to find the year when the purchasing power of a 2012 dollar is expected to reach $0.48. 50. China's shale-natural gas The function that models the growth in the number of billions of cubic feet of shale-natural gas in China, with x as the number of years after 2010, is y = 0.0117(1.75) Find the year when the number of cubic feet is projected to be 2.5 billion by solving algebraically (Source: Sanford C. Bernstein). 51. Supply Suppose the supply of x units of a product at price p dollars per unit is given by p= 10 + 5 In (3x + 1)
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