Concept explainers
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
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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. 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_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_forwardReminderRound all answers to two decimal places unless otherwise indicated. InflationDuring a period of high inflation, a political leader was up for re-election. Inflation had been increasing during his administration, but he announced that the rate of increase of inflation was decreasing. Draw a graph of inflation versus time that illustrates this situation. Would this announcement convince you that economic conditions were improving?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Mittenwalde Is Rich In 1562, the town of Mittenwalde lent Berlin 400 guilders repayable at 6 per year. The debt has yet to be repaid. a.Assuming that interest is compounded annually, how much in trillions of guilders did Berlin owe to Mittenwalde in 2012? b.We take the value of a 1562 guilder to be about 0.45 dollar. In trillions of dollars, how much was Berlins debt in 2012? c.Do you think it likely that Mittenwalde will ever collect from Berlin?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Account Growth The table below shows the balance B in a savings account, in dollars, in terms of time t, measured as the number of years since the initial deposit was made. Time t Balance B 0 125.00 1 131.25 2 137.81 3 144.70 4 151.94 a. Was the yearly interest rate constant over the first 4 years? If so, explain why and find that rate. If not, explain why not. Round the ratios to two decimal places. b. Estimate B(2.75) and explain in practical terms what your answer means. Assume that interest is compounded and deposited continuously.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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. WendysAccording to a report in The Wall Street Journal, Wendys revenue fell 3.2 to 489.5million in the second quarter of 2015. That represents a quarterly decay rate, as a decimal of e0.0325. Let R(t) denote Wendys revenue, in millions of dollars, t quarters after the second quarter of 2015. Suppose that revenue continues to fall at this same rate. a.Write the equation of change for Wendys revenue. b.Find a formula that gives Wendys revenue t quarters after the second quarter of 2015.arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Real Estate Sales A real estate agency has fixed monthly costs associated with rent, staff salaries, utilities, and supplies. It earns its money by taking a percentage commission on total real estate sales. During the month of July, the agency had total sales of 832,000 and showed a net income after paying fixed costs of 15,704. In August, total sales were 326,000 with a net income of only 532. a. Use a formula to express net income as a linear function of total sales. Be sure to identify what the letters that you use mean. b. Plot the graph of net income and identify the slope and vertical intercept. c. What are the real estate agencys fixed monthly costs? d. What percentage commission does the agency take on the sale of a home? e. Find the horizontal intercept and explain what this number means to the real estate agency.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_forward
- Reminder 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_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_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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning