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Reminder Round all answers to two decimal places unless otherwise indicated.
How Fast Do Exponential Functions Grow? At age
Retirement Option 1 When you retire, you will be paid a lump sum of
Retirement Option 2 When you start to work, the company will deposit
Which retirement option is more favorable to you if you retire at age
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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. 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_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_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
- 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. 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. 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. 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_forwardReminder Round all answers to two decimal places unless otherwise indicated. Theater Production Data from the Statistical Abstract of the United States show that in 1995, there were 56.61 thousand performances in the United States by nonprofit professional theaters. From 1995 through 2007, this number increased on average by about 10 each year. a.Let P denote the number of performances, in thousands, and let t denote the time in years since 1995. Make an exponential model for P versus t. b.How many performances by non-profit professionals theaters does your model give for 2007? The actual number was 197 thousand.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_forward
- Reminder 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. Growth of Bacteria The organism E. coli is a common bacterium. Under certain conditions, it undergoes cell division approximately each 20minutes. During cell division, each cell divides into two cells. a.Explain why the number of E. coli cells present is an exponential function of time. b.What is the hourly growth factor for E. coli? c.Express the population N of E. coli as an exponential function of time t measured in hours. Use N0 to denote the initial population. d.How long will it take a population of E. coli to triple in size?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Radioactive Iodine Iodine-131 is a radioactive form of iodine. After the crisis at a Japanese nuclear power plant in March 2011, elevated levels of this substance were detected thousands of miles away from Japan. Iodine-131 has a half-life of 8days. What is the daily decay factor for this substance?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning