171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 MAT 171 Precalculus Algebra 5.6 Applications and Models: Growth and Decay; and Compound Interest Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions exponential growth • Solve applied problems involving and decay. compound interest. • Solve applied problems involving 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest • Find models involving exponential functions and logarithmic functions. Population Growth The function P(t) = P0 ekt, k > 0 can model many kinds of population growths. Click globe on left to see TI Calculator tutorials at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm In this function: P0 = population at time 0, P(t) = population after time t, t = amount of time, k = exponential growth rate. The growth rate unit must be the same as the time unit. Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. There are two Interactive Figures for this section. (1) Exponential Growth Models (2) Graphs of Logistic Functions Use this GeoGebra program to calculate Exponential Growth and Exponential Decay. http://cfcc.edu/mathlab/geogebra/growth_decay.html Nov 2212:55 PM Nov 2212:55 PM Example (continued) Population Growth Graph Solution: a) At t = 0 (2006), the population was about 1.314 billion. We substitute 1.314 for P0 and 0.006 for k to obtain the exponential growth function. P(t) = 1.314e0.006t b) c) In 2010, t = 4. To find the population in 2010 we substitute 4 for t: Example In 2006, the population of China was about 1.314 billon, and the exponential growth rate was 0.6% per year. P(4) = 1.314e0.006(4) = 1.314e0.024 a) Find the exponential growth function. b) Graph the exponential growth function. c) Estimate the population in 2010. d) After how long will the population be double what it was in 2006? ≈ 1.346 Nov 2212:55 PM The population will be approximately 1.346 billion in 2010. The graph also displays this value. Nov 2212:55 PM 1 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 Example (continued) d) We are looking for the doubling time; T such that P(T) = 2 • 1.314 = 2.628. Solve Interest Compound Continuously The function P(t) = P0ekt can be used to calculate interest that is compounded continuously. The population of China will be double what it was in 2006 about 115.5 years after 2006. d) Using the Intersect method we graph and find the first coordinate of their point of intersection. In this function: P0 = amount of money invested, P(t) = balance of the account after t years, t = years, k = interest rate compounded continuously. Suppose that $2000 is invested at interest rate k, compounded continuously, and grows to $2504.65 after 5 years. a. What is the interest rate? b. Find the exponential growth function. c. What will the balance be after 10 years? d. After how long will the $2000 have doubled? The population of China will be double that of 2006 about 115.5 years after 2006. Nov 2212:55 PM Nov 2212:55 PM Example (continued) Solution: a. At t = 0, P(0) = P0 = $2000. Thus the exponential growth function is P(t) = 2000ekt . We know that P(5) = $2504.65. Substitute and solve for k: Growth Rate and Doubling Time The growth rate k and doubling time T are related by The interest rate is about 0.045 or 4.5%. b. The exponential growth function is P(t) = 2000 e 0.045 t kT = ln 2 or or Note that the relationship between k and T does not depend on P0 . . c. The balance after 10 years is Example The population of the world is now doubling every 60.8 yr. What is the exponential growth rate? d. To find the doubling time T, we set P(T) = 2 • P0= 2 • $2000 = $4000 and solve for T. Solution: The growth rate of the world population is about 1.14% per year. Thus the orginal investment of $2000 will double in about 15.4 yr. Nov 2212:55 PM Nov 2212:55 PM 2 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 Models of Limited Growth In previous examples, we have modeled population growth. However, in some populations, there can be factors that prevent a population from exceeding some limiting value. One model of such growth is which is called a logistic function. This function increases toward a limiting value a as t approaches infinity. Thus, y = a is the horizontal asymptote of the graph. Exponential Decay Decay, or decline, of a population is represented by the function P(t) = P0e−kt , k > 0. In this function: P0 = initial amount of the substance (at time t = 0), P(t) = amount of the substance left after time, t = time, k = decay rate. The halflife is the amount of time it takes for a substance to decay to half of the original amount. Models of Limited Growth Graph Growth Graph Nov 2212:55 PM Decay Graph Nov 2212:55 PM Example (continued) Decay Rate and HalfLife Solution: First find k when the halflife T is 5750 yr: The decay rate k and the halflife T are related by kT = ln 2 or or Note that the relationship between decay rate and halflife is the same as that between growth rate and doubling time. Example Carbon Dating . The radioactive element carbon14 has a halflife of 5750 years. The percentage of carbon14 present in the remains of organic matter can be used to determine the age of that organic matter. Archaeologists discovered that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of its carbon14 at the time it was found. How old was the linen wrapping? If the linen wrapping lost 22.3% of its carbon14 from the initial amount P0, then 77.7% is the amount present. To find the age t of the wrapping, solve for t: The linen wrapping on the Dead Sea Scrolls was about 2103 years old when it was found. Nov 2212:55 PM Nov 2212:55 PM 3 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 Exponential Curve Fitting We have added several new functions that can be considered when we fit curves to data. Example The number of U.S. communities using surveillance cameras at intersections has greatly increased in recent years, as show in the table. Logarithmic Curve Fitting Logarithmic Logistic a. Use a graphing calculator to fit an exponential function to the data. b. Graph the function with the scatter plot of the data. c. Estimate the number of U.S. communities using surveillance cameras at intersections in 2010. Nov 2212:55 PM Nov 2212:55 PM Example (continued) Solution: a. Fit an equation of the type y = a • bx, where x is the number of years since 1999. Enter the data . . . 463/3a, c. Population Growth following table. . Complete the The correlation coefficient is close to 1, indicating the exponential function fits the data well. b. Here’s the graph of the function with the scatter plot. c. Using the VALUE feature in the CALC menu, we evaluate the function for x = 11 (2010 – 1999 = 11), and estimate the number of communities using surveillance cameras at intersections in 2010 to be about 651. Nov 2212:55 PM Nov 221:27 PM 4 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 463/3b, d. Population Growth following table. . Complete the Nov 221:27 PM 463/3h, j. Population Growth following table. 463/3e, f. Population Growth following table. . Complete the Nov 221:27 PM . Complete the 464/8a. Interest Compounded Continuously Complete the following table. Nov 221:27 PM . Nov 221:27 PM 5 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 464/8b. Interest Compounded Continuously Complete the following table. . Nov 221:27 PM 464/8f. Interest Compounded Continuously Complete the following table. Nov 221:27 PM 464/8c. Interest Compounded Continuously Complete the following table. . Nov 221:27 PM . 465/11a. Radioactive Decay following table. . Complete the Nov 221:27 PM 6 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 465/11c. Radioactive Decay following table. . Complete the 465/11d. Radioactive Decay following table. Nov 221:27 PM 465/11e. Radioactive Decay following table. . Complete the Nov 221:27 PM . Complete the 467. In Exercises 2328 , determine which, if any, of these functions might be used as a model for the data in the scatterplot. a) Quadratic, f(x) = ax 2 + bx + c b) Polynomial, not quadratic c) Exponential, f(x) = ab x , or Poe kx , k > 0 d) Exponential, f(x) = ab x , or Poe kx , k > 0 e) Logarithmic, f(x) = a + b ln x f) Logistic, 23. 24. 25. 26. Nov 221:27 PM Nov 221:42 PM 7 171S5.6q Applications and Models: Growth and Decay; ?and Compound Interest April 24, 2013 468/30. Forgetting . In an economics class, students were given a final exam at the end of the course. Then they were retested with an equivalent test at subsequent time intervals. Their scores after time x, in months, are listed in the table below. 468/31. Unoccupied Homes in a Census. After doortodoor canvassing, census workers determine the number of unoccupied homes. The table below lists the number of homes, in millions, that were unoccupied in four census years. a) Use a graphing calculator to fit a logarithmic function y = a + b ln x to the data. b) Use the function to predict test scores after 8, 10, 24, and 36 months. c) After how long will the test scores fall below 82%? a) Use a graphing calculator to fit an exponential function to the data, where x is the number of years after 1980. b) Graph the function found in part (a). c) Use the function found in part (a) to project the number of unoccupied homes in the 2020 Census. f(x) = a + b ln x Nov 221:57 PM 468/32. Older Americans in Debt. The debt among older Americans has been exponentially increasing in recent years. The table below lists, for selected years, the average debt per household, where the primary head is age 55 or older. a) Use a graphing calculator to fit the data with an exponential function, where x is the number of years since 2000. b) Graph the function found in part (a). c) Use the function in part (a) to estimate the debt in 2014. d) In what year is average debt among older Americans expected to exceed $140,000? Nov 221:57 PM Nov 221:57 PM 469/34. Effect of Advertising. A company introduced a new software product on a trial run in a city. They advertised the product on television and found the following data regarding the percent P of people who bought the product after x ads were run. a) Use a graphing calculator to fit a logistic function to the data. b) What percent of people bought the product when 55 ads were run? 100 ads? c) Find the horizontal asymptote for the graph. Interpret the asymptote in terms of the advertising situation. Nov 221:57 PM 8
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