509 4.6 Applications and Models of Exponential Growth and Decay 30. Growth of an Account If Russ (see Exercise 29) chooses the plan with continuous compounding, how long, to the nearest hundredth of a year, will it take for his $60,000 to grow to $70,000? 31. Doubling Time Find the doubling time, to the nearest hundredth of a year, of an investment earning 2.5% interest if interest is compounded continuously. 32. Doubling Time If interest is compounded continuously and the interest rate is tripled, what effect will this have on the time required for an investment to double? 33. Growth of an Account How long, to the nearest hundredth of a year, will it take an investment to triple if interest is compounded continuously at 3%? 34. Growth of an Account Use the Table feature of a graphing calculator to find how long it will take $1500 invested at 2.75% compounded daily to triple in value. Zoom in on the solution by systematically decreasing the increment for x. Find the answer to the nearest day. (Do this by eventually letting the increment of x equal 1 365. The decimal part of the solution can be multiplied by 365 to determine the number of days greater than the nearest year. For example, if the solution is determined to be 16.2027 yr, then multiply 0.2027 by 365 to get 73.9855. The solution is then, to the nearest day, 16 yr, 74 days.) Confirm the answer algebraically. Social Sciences (Exercises 35–44) 35. Legislative Turnover The turnover of legislators is a problem of interest to political scientists. It was found that one model of legislative turnover in a particular body was M1t2 = 434e-0.08t, where M1t2 represents the number of continuously serving members at time t. Here, t = 0 represents 2010, t = 1 represents 2011, and so on. Use this model to approximate the number of continuously serving members in each year. (a) 2014 (b) 2018 (c) 2024 36. Legislative Turnover Use the model in Exercise 35 to determine the year in which the number of continuously serving members was 338. 37. Population Growth In 2000 India’s population reached 1 billion, and it is projected to be 1.4 billion in 2025. (Data from U.S. Census Bureau.) (a) Find values for P0 and a so that P1x2 = P0ax-2000 models the population of India in year x. Round a to five decimal places. (b) Predict India’s population in 2020 to the nearest tenth of a billion. (c) In what year is India’s population expected to reach 1.5 billion? 38. Population Decline A midwestern city finds its residents moving to the suburbs. Its population is declining according to the function P1t2 = P0e-0.04t, where t is time measured in years and P0 is the population at time t = 0. Assume that P0 = 1,000,000. (a) Find the population at time t = 1 to the nearest thousand. (b) How long, to the nearest tenth of a year, will it take for the population to decline to 750,000? (c) How long, to the nearest tenth of a year, will it take for the population to decline to half the initial number?
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