Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Theater Production Data from the Statistical Abstract of the United States show that in
a. Let P denote the number of performances, in thousands, and let
b. How many performances by non-profit professionals theaters does your model give for
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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. 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. 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 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. Gray Wolves in MichiganGray wolves recolonized in the Upper Peninsula of Michigan beginning in 1990. Their population has been documented as shown in the following table. 56 a.Explain why one expect an exponential model to be appropriate for these data. b.Find an exponential model for the data given. Year Wolves 1990 6 1991 17 1992 21 1993 30 1994 57 1995 80 1996 116 1997 112 1998 140 1999 174 2000 216 c.Graph the data and the exponential model. Would it be better to use a piecewise-defined function? d.Find an exponential model for 1990 through 1996 and another for 1997 through 2000. e.Write a formula for the number of wolves as a piecewise-defined function using the two exponential models. Is the combined model a better fit?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_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. 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. 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. 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_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. Gold Prices During the period from 2003 through 2010, gold prices doubled every 3years approximately. a.What was the yearly growth factor for the price of gold during this period? b.Explain in practical terms the meaning of the growth factor you found in part a.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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning