Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Unit Conversion with Exponential Growth The exponential function
a. What is the yearly growth factor? Find a formula that gives the population
b. What is the century growth factor? Find a formula that gives the U.S. population
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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. 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_forwardReminder Round all answers to two decimal places unless otherwise indicated. Unit conversion with Exponential Growth The exponential growth function N=35001.77d, where d is measured in decades, gives the number of individuals in a certain population. a.Calculate N(1.5) and explain what your answer means. b.What is the percentage growth rate per decade? c.What is the yearly growth factor rounded to three decimal places? What is the yearly percentage growth rate? d.What is the growth factor rounded to two decimal places for a century? What is the percentage growth rate per century?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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. How Fast Do Exponential Functions Grow? At age 25, you start to work for a company and are offered two rather fanciful retirement options. Retirement Option 1 When you retire, you will be paid a lump sum of 25,000 for each year of service. Retirement Option 2 When you start to work, the company will deposit 10,000 into an account that pays a monthly interest rate of 1. When you retire, the account will be closed and the balance given to you. Which retirement option is more favorable to you if you retire at age 65? Which retirement option is more favorable if you retire at age 55?arrow_forwardReminder 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. 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. 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. 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 Decay The exponential function N=5000.68t, where t is measured in years, shows the amount, in grams, of a certain radioactive substance present. a. Calculate N(2) and explain what your answer means. b. What is the yearly percentage decay rate? c. What is the monthly decay factor rounded to three decimal places? What is the monthly percentage decay rate? d. What is the percentage decay rate per second? Note: For this calculation, you will need to use all the decimal places that your calculator can show.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. 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. The Beer-Lambert-Bouguer Law When light strikes the surface of a medium such as water or glass, its intensity decreases with depth. The Beer-Lambert-Bouguer law states that the percentage of decrease is the same for each additional unit of depth. In a certain lake, intensity decreases about 75 for each additional meter of depth. a. Explain why intensity I is an exponential function of depth d in meters. b. Use a formula to express intensity I as an exponential function of d. Use Io to denote the initial intensity. c. Explain in practical terms the meaning of Io. d. At what depth will the intensity of light be one-tenth of the intensity of light striking the surface?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning