Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Wages A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage
Time
|
Wages
|
|
|
|
|
|
|
|
|
a. By calculating ratios, show that the data in this table are exponential. (Round the quotients to two decimal places)
b. What is the yearly growth factor for the data?
c. The worker can't remember what hourly wage he earned at the beginning of 2000. Assuming that
d. Find a formula giving an exponential model for
e. What percentage raise did the worker receive each year?
f. Given that prices increased by 26% over the decade of the 2000s your model to determine whether the workers wage increases kept pace with inflation.
Want to see the full answer?
Check out a sample textbook solutionChapter 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. 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. 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. 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. 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. 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. 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_forward
- 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. A Population with Given Per Capita Growth Rate A certain population has a yearly per capita growth rate of 2.3, and the initial value is 3 million. a.Use a formula to express the population as an exponential function. b.Express the population after 4 years using function notation, and then calculate that value.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Defense SpendingData about recent federal defense spending are given in the accompanying Statistical Abstract of the United States table. Here t denotes the time, in years, since 1990 and D denotes federal defense spending, in billions of dollars. a.Calculate the average yearly rate of change in defense spending from 1990 to 1995. b.Use your answer from part a to estimate D(3), and explain what it means. t= Years since 1990 D= Spending billions of dollars 0 328.4 5 310.0 10 341.5 15 565.5 20 843.8 c.Calculate the average yearly rate of change in defense spending from 2005 to 2010. d.Use your answer form part c to estimate the value of D(22).arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Sales Income The following table shows the net monthly income N for a real estate agency as a function of the monthly real estate sales s, both measured in dollars. s=Sales N=Netincome 450,000 4000 500,000 5500 550,000 7000 600,000 8500 a. Make a table showing, for each of the intervals in the tax table above, the average rate of change in N. What pattern do you see? b. Use the average rate of change to estimate the net monthly income for monthly real estate sales of 520,000. In light of your answer to part a, how confident are you that your estimate is an accurate representation of the actual income? c. Would you expect N to have a limiting value? Be sure to explain your reasoning.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Data That Are Not Exponential Show that the following data are not exponential. t h(t) t h(t) 0 4.9 3 200.2 1 26.6 4 352.1 2 91.7 5 547.4arrow_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
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning