Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
A Weight-Gain Program A woman who has recovered from a serious illness begins a diet regimen designed to get her back to a healthy weight. She currently weighs
a. Find a formula for an exponential function that gives the woman’s weight
b. How long will it be before she reaches her normal weight of
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. Mobile Phones According to one source, the amount of data passing through mobile phone networks doubles each year. a. Explain why the amount of data passing through mobile phone networks is as exponential function of time. b. Use D0 for the initial amount of data, and find a formula that gives the data D as an exponential function of the time t in years. c. If this trend continues, how long will it be before the amount of data is 100 times its initial value?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. U.S Investment Abroad In 1980, direct U.S. business investment abroad was about 13.5 billion dollars. From 1980 through 2010, that investment grew at an average annual rate of 11.24. a.Make an exponential model that shows the U.S. direct investment aboard A, in billions of dollars, t years after 1980. b.From 1980, how long did it take for U.S. investments abroad to double? c.According to the model, how long would it take from 2010 for investments abroad to double the level present in 2010?arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. Inflation An economist tracks the price of a certain item at the beginning of several years and compiles the following table. Years Price, in dollars 2013 265.50 2014 273.47 2015 281.67 2016 290.12 a. Show that the price is growing as an exponential function. b. Find an exponential model for the data. c. At the beginning of some year, the price will surpass 325. Use your model to determine which year.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. A Savings Account You initially invested 500 in a saving account that pays a yearly interest rate of 4. a. Write a formula for an exponential function giving the balance in your account as a function of the time since your initial investment. b. What monthly interest rate best represents this account? Round your answer to three decimal places. c. Calculate the decade growth factor. d. Use the formula you found in part a to determine how long it will take for the account to reach 740. Explain how this is consistent with your answer to part c.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Internet Domain Hosts Often new technology spreads exponentially. Between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in the preceding year. In 1995, the number of hosts was 8.2million. a Explain why the number of hosts is an exponential function of time. b Find the formula for the exponential function that gives the number N of hosts, in millions, as a function of the time t in years since 1995. c According to this model, in what year did the number of hosts reach 24million.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. A Violin String A violin string is stopped so that the resulting string length makes a desired musical note. In order to make the next higher note, the string must be shortened using a factor of 2 or about 0.944. That is, the current length is multiplied by 0.944. The length of an unstopped string is 32centimeters. a.Find a formula for an exponential function that gives the length L, in centimeters, of a string that is stopped to make a tone n notes higher than the unstopped string. b.One of the unstopped strings makes an A note. To what length must the string be stopped in order to make C, which is 4 notes higher?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. APR and APY Recall that financial institutions sometimes report the annual interest rate that they offer on investments as the APR, often called the nominal interest rate. To indicate how an investment will actually grow, they advertise the annual percentage yield, or APY. In mathematical terms, this is the yearly percentage growth rate for the exponential function that models the account balance. In this exercise and the next, we study the relationship between the APR and the APY. We assume that the APR is 10 or 0.1 as a decimal. To determine the APY when we know the APR, we need to know how often interest is compounded. For example, suppose for the moment that interest is compounded twice a year. Then to say that the APR is 10 means that in half a year, the balance grows by 102 or 5. In other words, the 12-year age growth rate is 0.12 as a decimal. Thus, the 12-year growth factor is 1+0.12. To find the yearly growth factor, we need to perform a unit conversion: One year is 2 half-year periods, so the yearly growth factor is (1+0.12)2, or 1.1025. a. What is the yearly growth factor if interest is compounded four times a year? b. Assume that interest is compounded n times each year. Explain why the formula for the yearly growth factor is (1+0.1n)n. c. What is the yearly growth factor if interest is compounded daily? Give your answer to four decimal places/arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Gray Wolves in Idaho The report cited in Example 4.6 tells us that in 2009, there were 870 gray wolves in Idaho, but that the population declined by 19 that year. For purposes of this problem, we assume that this 19 annual rate of decrease continues. a. Find an exponential model that gives the wolf population W as function of the time t in years since 2009. b. It is expected that the wolf population cannot recover if there are fewer than 20 individuals. How long must this rate of decline continue for the wolf population to reach 20?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Long-Term Population Growth Although exponential growth can often be used to model population growth accurately for some periods of time, there are inevitably, in the long term, limiting factors that make purely exponential models inaccurate. From 1790 to 1860, the U.S. population could be modeled by N=3.931.03tmillion people, where t is the time in years since 1790. If this exponential growth rate had continued until today, what would be the population of the United States have been in 2015? Compare your answer with the actual population of the United States in 2015, which was about 323million.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Unit Conversion with Exponential Growth The exponential function N=3.931.34d gives the approximate U.S. population, in millions, ddecades after 1790. The formula is valid only up to 1860. a.What is the yearly growth factor? Find a formula that gives the population yyears after 1790. b.What is the century growth factor? Find a formula that gives the U.S. population ccenturies after 1790. Assume that the original formula is valid over several centuries.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Doubling Time The current world population is about 7.3billion. Under current conditions, the population is growing exponentially, with a yearly growth factor of 1.011. In parts b and c, round your answers to the nearest year. a.Find a formula that gives the world population N, in billions, after tyears. b.How long will it take for the population to double? c.How long after doubling will it take for the population to double again?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Unit Conversion with Exponential Decay The exponential function P=3000.86m gives the amount in parts per million of PCBs in a contaminated site mmonths after a cleanup process begins. a.What is the weekly decay factor? Find a formula that gives the amount P in parts per million wweeks after cleanup begins. Assume that there are four weeks in each month. b.What is the yearly decay factor? Find a formula that gives the amount P in parts per million yyears after cleanup begins.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning