Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Doubling Time The current world population is about
a. Find a formula that gives the world population
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?
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Chapter 4 Solutions
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_forwardReminder 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_forward
- 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 104pounds. She hopes to multiply her wait by 1.03 each week. a.Find a formula for an exponential function that gives the womans weight w, in pounds, after tweeks on the regimen. b.How long will it be before she reaches her normal weight of 135pounds?arrow_forwardReminder 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_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 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_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. 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
- Reminder 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_forwardReminder Round all answers to two decimal places unless otherwise indicated. An Investment You have invested money in a savings account that pays a fixed monthly interest on the account balance. The following table shows the account balance over the first 5 months. Time, in months Saving balance 0 1750.00 1 1771.00 2 1792.25 3 1813.76 4 1835.52 5 1857.55 a. How much money was originally invested? b. Show that the data are exponential and find an exponential model for the account balance. c. What is the monthly interest rate? d. What is the yearly interest rate? e. Suppose that you made this investment on the occasion of the birth of your daughter. Your plan is to leave the money in the account until she starts college at age 18. How large a college fund will she have? f. How long does it take your money to double in value? How much longer does it take it to double in value again?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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning