Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Growth Rate An animal grows according to the formula
Here
a. Draw a graph of length versus age. Include ages up to
b. Explain in practical terms what
c. How old is the animal when it is
d. Explain in practical terms what the concavity of the graph means.
e. Use the formula to express the age as a function of the length.
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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. The pH Scale Acidity of a solution is determined by the concentration H of hydrogen ions in the solution measured in moles per liter of solution. Chemists use the negative of the logarithm of the concentration of hydrogen ions to define the pH scale: pH=logH Lower pH values indicate a more acidic solution. a.Normal rain has a pH value of 5.6. Rain in the eastern United States often has a pH level of 3.8. How much more acidic is this than normal rain? b.If the pH of water in lake falls below a value of 5, fish often fail to reproduce. How much more acidic is this than normal water with a pH of 5.6?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. Logistic Model A population grows according to the logistic model. The r value is 0.02 and the environmental carrying capacity is 2500. Write the logistic equation satisfied by the population if N(0)=100.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. 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. 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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Doubling Time If an investment earns an APR of r, as a decimal, compounded annually, then the time D, in years, required for the investment to double in value is given by D=log2log(1+r). a.Find the doubling time for an investment subject to an APR of 5 if interest is compounded annually. b.Plot the graph of the doubling time D versus the interest rate r, as a decimal. Use a horizontal span of 0 to 0.1. c.Does a small change in the interest rate have a greater effect on the doubling time if interest rates are low or if they are high?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Minimum Monthly PaymentSuppose you have a balance of B dollars on credit card.You choose to stop charging and pay off the card, making only minimum monthly payments.If your card charges an APR of r, as a decimal, and requires a minimum monthly payment of 5 of the balance, then the time T, in months, required to reduce your balance to 100 is given by T=2logBlog(0.95(1+r/12)). Suppose your current balance is 8000. a.How long will it take to reduce your balance to 100 if the APR for your card is 25? Report your answer to the nearest whole month. b.Plot the graph of T versus r. Use a horizontal span of 0 to 0.3. c.Does a larger APR mean a longer or a shorter time to reduce the balance to 100?arrow_forwardReminder 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_forward
- ReminderRound 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. 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. 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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning