Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Unit Conversion with Exponential Decay The exponential function
a. What is the weekly decay factor? Find a formula that gives the amount
b. What is the yearly decay factor? Find a formula that gives the amount
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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. Unit Conversion with Exponential Growth The exponential function N=3.931.34d gives the approximate U.S. population, in millions, ddecades after 1790. The formula is valid only up to 1860. a.What is the yearly growth factor? Find a formula that gives the population yyears after 1790. b.What is the century growth factor? Find a formula that gives the U.S. population ccenturies after 1790. Assume that the original formula is valid over several centuries.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Mobile Phones According to one source, the amount of data passing through mobile phone networks doubles each year. a. Explain why the amount of data passing through mobile phone networks is as exponential function of time. b. Use D0 for the initial amount of data, and find a formula that gives the data D as an exponential function of the time t in years. c. If this trend continues, how long will it be before the amount of data is 100 times its initial value?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. An Investment You have invested money in a savings account that pays a fixed monthly interest on the account balance. The following table shows the account balance over the first 5 months. Time, in months Saving balance 0 1750.00 1 1771.00 2 1792.25 3 1813.76 4 1835.52 5 1857.55 a. How much money was originally invested? b. Show that the data are exponential and find an exponential model for the account balance. c. What is the monthly interest rate? d. What is the yearly interest rate? e. Suppose that you made this investment on the occasion of the birth of your daughter. Your plan is to leave the money in the account until she starts college at age 18. How large a college fund will she have? f. How long does it take your money to double in value? How much longer does it take it to double in value again?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. A Violin String A violin string is stopped so that the resulting string length makes a desired musical note. In order to make the next higher note, the string must be shortened using a factor of 2 or about 0.944. That is, the current length is multiplied by 0.944. The length of an unstopped string is 32centimeters. a.Find a formula for an exponential function that gives the length L, in centimeters, of a string that is stopped to make a tone n notes higher than the unstopped string. b.One of the unstopped strings makes an A note. To what length must the string be stopped in order to make C, which is 4 notes higher?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_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. Unit Conversion with Exponential Decay The exponential function N=5000.68t, where t is measured in years, shows the amount, in grams, of a certain radioactive substance present. a. Calculate N(2) and explain what your answer means. b. What is the yearly percentage decay rate? c. What is the monthly decay factor rounded to three decimal places? What is the monthly percentage decay rate? d. What is the percentage decay rate per second? Note: For this calculation, you will need to use all the decimal places that your calculator can show.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. The Beer-Lambert-Bouguer Law When light strikes the surface of a medium such as water or glass, its intensity decreases with depth. The Beer-Lambert-Bouguer law states that the percentage of decrease is the same for each additional unit of depth. In a certain lake, intensity decreases about 75 for each additional meter of depth. a. Explain why intensity I is an exponential function of depth d in meters. b. Use a formula to express intensity I as an exponential function of d. Use Io to denote the initial intensity. c. Explain in practical terms the meaning of Io. d. At what depth will the intensity of light be one-tenth of the intensity of light striking the surface?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. A Weight-Gain Program A woman who has recovered from a serious illness begins a diet regimen designed to get her back to a healthy weight. She currently weighs 104pounds. She hopes to multiply her wait by 1.03 each week. a.Find a formula for an exponential function that gives the womans weight w, in pounds, after tweeks on the regimen. b.How long will it be before she reaches her normal weight of 135pounds?arrow_forward
- Reminder 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_forwardReminderRound 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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning