Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Mittenwalde Is Rich In
a. Assuming that interest is compounded annually, how much (in trillions of guilders) did Berlin owe to Mittenwalde in
b. We take the value of a
c. Do you think it likely that Mittenwalde will ever collect from Berlin?
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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. Inflation The yearly inflation rate tells the percentage by which prices increase. For example, from 1990 through 2000, the inflation rate in the United States remained stable at amount 3 per year. In 1990, an individual retired on a fixed income of 36,000 per year. Assuming that the inflation rate remains at 3, determine how long it will take for the retirement income to deflate to half its 1990 value. Note: To say that retirement income has deflated to half its 1990 value means that prices have doubled.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. 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. World Copper Production World production of copper, in millions of tons per year, from 1900 to 2000 is given by C=0.51.033t, where t is the time in years since 1900. a.What production level does this model give for the year 2000? b.If this model were extended to 2025, how could you use your knowledge of copper production in 2024 to estimate copper production in 2025?arrow_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. 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. 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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Small Business Loan After t years, a small business owes B dollars on a loan. The balance draws a continuous compounding rate of 6 per year, and payments of 5000 per year are made to the lending institution. Then the equation of change for the account balance is given by dBdt=0.06B5000 a. Find the equilibrium solution. b. Explain what is happening at the equilibrium solution in practical terms. c. If the small business expects to pay off the loan eventually, should B be larger or smaller than the equilibrium solution?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. What if Interest is Compounded More Often than Monthly?Some lending institutions compound interest daily or even continuously. The term continuous compounding is used when interest is being added as often as possible-that is, at each instant in time. The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part 1 of Example 1.2, we showed that if you borrow 7800 from an institution that compounds monthly at a monthly interest rate of 0.67 for an APR of 8.04 , then in order to pay off the note in 48months, you have to make a monthly payment of 190.57. a.Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why. b.Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning. c.When interest is compounded continuously, you can calculate your monthly payment M=M(P,r,t) in dollars, for a loan of Pdollars to be paid off over t months using M=P(er1)1ert, where r=APR/12 if the APR is written in the decimal form. Use this formula to calculate the monthly payment on a loan of 7800 to be paid off over 48months with an APR of 8.04. How does this answer compare the result in Example 1.2?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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning