Reminder Round all answers to two decimal places unless otherwise indicated.
Wendy’s According to a report in The Wall Street Journal, Wendy’s revenue fell
a. Write the equation of change for Wendy’s revenue.
b. Find a formula that gives Wendy’s revenue
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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 half life of 239U Uranium-239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of 239U was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below. Time, in minutes Grams remaining 0 1 1 0.971 2 0.943 3 0.916 4 0.889 5 0.863 a. Show that these are exponential data and find an exponential model For this problem, round all your answers to three decimal places. b. What is the percentage decay rate each minute? What does this number mean in practical terms? c. Use functional notation to express the amount remaining after 10 minutes and then calculate the value. d. What is the half life of 239U?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_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
- Reminder 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_forwardReminder 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. Quarterly Pine Pulpwood PricesIn southwest Georgia, the average pine pulpwood prices vary predictably over the course of the year, primarily because of weather. Prices in 2009 followed this pattern. At the beginning of the first quarter, the average price P was 9 per ton. During the first quarter, prices declined steadily to 8 per ton, then remained steady at 8 per ton through the end of the third quarter. During the fourth quarter, prices increased steadily from 8 to 10 per ton. a.Sketch a graph of pulpwood prices as a function of the quarter in the year. b.What formula for price P as a function of t, the quarter, describes the price from the beginning of the year through the first quarter? c.What formula for price P as a function of t, the quarter, describes the price from the first to the third quarter? d.What formula for price P as a function of t, the quarter, describes the price from the third to the fourth quarter? e.Write a formula for price P throughout the year as a piecewise-defined function of t, the quarter.arrow_forward
- Round all answers to two decimal places. InflationDuring a certain period, the price of a bag of groceries grows by about 3 each year. Which regression model should be used to approximate the price of a bag of groceries as a function of time: linear or exponential?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. t is measured in thousands of years, and C=C(t) is the amount, in grams, of carbon-14 remaining. Carbon-14 unstable radioactive t=Thousandofyears C=Gramsremaining 0 5 5 2.73 10 1.49 15 0.81 20 0.44 a. What is the average yearly rate of change of carbon-14 during the first 5000 years? b. How many grams of carbon-14 would you expect to find remaining after 1236 years? c. What would you expect to be the limiting value of C?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Marriage Length In this exercise, we consider data from the Statistical Abstract of the United States on the fraction of women married for the first time in 1960 whose marriage reached a given anniversary number. The data show that the fraction of women who reached their fifth anniversary was 0.928. After that, for each one-year anniversary number, the fraction reaching that number drops by about 2. These data describe constant percentage change, so it is reasonable to model the fraction M as an exponential function of the number n of anniversaries since fifth. a.What is the yearly decay factor for the exponential model? b.Find an exponential model for M as a function of n. c.According to your model, what fraction of women married for the first time in 1960 celebrated their 40th anniversary? Take n=35. Round your answer to three decimal places. The actual fraction is 0.449 or 44.9.arrow_forward
- Reminder 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. 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. Moores Law The speed of a computer chip is closely related to the number of transistors on the chip, and the number of transistors on a chip has increased with time in a remarkably consistent way. In fact, in the year 1965, Dr. Gordon E. Moore now chairman emeritus of Intel Corporation observed a trend and predicted that it would continue for a time. His observation, now known as Moores law, is that every two years or so a chip is introduced with double the number of transistors of its fastest predecessor. This law can be restated in the following way: If time increases by 1year, then the number of transistors is multiplied by 100.15.More generally, the rule is that if time increases by tyears, then the number of transistors is multiplied by 100.15t.For example, after 8years, the number of transistors is multiplied by 100.158, or about 16. The 6th generation Core processor was released by Intel Corporation in the year 2015. a.If a chip were introduced in the year 2022, how many times the transistors of the 6th generation Core would you expect it to have? Round your answer to the nearest whole number. b.The limit of conventional computing will be reached when the size of a transistors on a chip will be 200 times that of the 6th generation Core. When, according to Moores law, will that limit be reached? c.Even for unconventional computing, the law of physics impose a limit on the speed of computation. The fastest speed possible corresponds to having about 1040 times the number of transistors as on the 6th generation Core. Assume that Moores law will continue to be valid even for unconventional computing, and determine when this limit will be reached. Round your answer to the nearest century.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning