Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Gray Wolves in Idaho The report cited in Example
a. Find an exponential model that gives the wolf population
b. It is expected that the wolf population cannot recover if there are fewer than
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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. 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_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. 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. 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. 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_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. 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. Continuous CompoundingThis is a continuation of Exercise 22. In this exercise, we examine the relationship between APR and the APY when interest is compounded continuously-in other words, at every instant. We will see by means of an example that the relationship is Yearlygrowthfactor=eAPR,(4.1) and so APY=eAPR1(4.2) if both the APR and the APY are in decimal form and interest is compounded continuously. Assume that the APR is 10, or 0.1 as a decimal. a.The yearly growth factor for continuous compounding is just the limiting value of the function given by the formula in part b of Exercise 22. Find that limiting value to four decimal places. b.Compute eAPR with an APR of 0.1 as a decimal. c.Use your answers to parts a and b to verify that Equation 4.1 holds in the case where the APR is 10. Note: On the basis of part a, one conclusion is that there is a limit to the increase in the yearly growth factor and hence in the APY as the number of compounding periods increases. We might have expected the APY to increase without limit for more and more frequent compounding. 22. APR and APYRecall 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 percentage 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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. A Diet An overweight man makes lifestyle changes in order to lose weight. He currently weighs 260pounds, and he has set a target weight of 200pounds. Each month the difference D, in pounds, between his current weight and his target weight decreases by 10. a. Make an exponential model of D versus the time t in months since the diet began. b. How long will it take for his weight to reach 210poundsarrow_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_forwardReminder Round all answers to two decimal places unless otherwise indicated. Population Growth A population of animals is growing exponentially, and an ecologist has made the following table of the population size, in thousands, at the start of the given year. Year Population, in thousands 2011 5.25 2012 5.51 2013 5.79 2014 6.04 2015 6.38 2016 6.70 Looking over the table, the ecologist realizes that one of the entries for population size is in error. Which entry is it, and what is the correct population? Round the ratios to two decimal places.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning