rig? ACTIVITY 5.6 b. Determine the population of Charlotte (in thousands) in 2002.

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1 ACTIVITY 5.6 POPULATION GROWTH 581 OBJECTIVES ACTIVITY 5.6 Population Growth 1, Determine annual growth or decay rate of an exponential function represented by a table of values or an equation. 2. Graph an exponential function having equation >> = a(l + r)*. According to the 2000 U.S. census, the city of Charlotte, North Carolina, had a population of approximately 541, a. Assuming that the population increases at a constant rate of 3.2%, determine the population of Charlotte (in thousands) in b. Determine the population of Charlotte (in thousands) in c. Divide the population in 2001 by the population in 2000, and record this ratio. d. Divide the population in 2002 by the population in 2001, and record this ratio. e. What do you notice about the ratios in parts c and d? What do these ratios represent? Linear functions represent quantities that change at a constant average rate (slope). Exponential functions represent quantities that change at a constant ratio, expressed as a percent. Example 1 Population growth, sales and advertising trends, compound interest, spread of disease, and concentration of a drug in the blood are examples of quantities that increase or decrease at a constant rate expressed as a percent. 2. Let t represent the number of years since 2000 (/ = 0 corresponds to 2000). Use the results from Problem 1 to complete the following table: t, Years (since 2000) P, population (in thousands) 541 rig? Charlotte jf"

2 582 CHAPTER 5 MODELING WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS Once you know the growth factor, b, and the initial value, a, you can write the exponential equation. In this situation, the initial value is 541, the population, in 1000s, in 2000 (t = 0). The growth factor is b = b. Write the exponential equation, P = a b', for the population of Charlotte. 3. a. Write the growth rate, r = 3.2%, as a decimal. b. Add 1 to the decimal form of the growth rate r. The growth factor, b, is determined from the growth rate, r, by writing r in decimal form and adding 1: b = 1 + r. Example 2 Determine the growth factor, b, for a growth rate of r = 8 %. r = 8% = 0.08, b = 1 + r = = 1.08 c. Solve the equation for the growth factor, b = 1 + r, for r. The growth rate, r, is determined from the growth factor, b, by subtracting 1 from b and writing the result in percent form. Example 3 Determine the growth rate, r, for a growth factor ofb = r = b - I = = = 5.4% The growth rate is 5.4% 4. a. Complete the following table: CALCULATION FOR POPULATION (in thousands) 541 (541)1.032 (541)(1.032)0.032) EXPONENTIAL FORM 541(1.032). 541( 1.032) 1 b. Use the pattern in the table in part a to help you write an equation for P, th population of Charlotte (in thousands), using r, the number of years sinc 2000, as the independent variable.

3 ACTIVITY 5.6 POPULATION GROWTH 583 The equation P = 541 (1.032)'has the general form P = P 0 (l + / )', where r is the annual growth rate, (1 + r) is the growth factor or the base, b, of the exponential function, t is the time in years, and P 0 is the initial value, the population when t = 0. : Example 4 a. Determine the growth factor and the growth rate of the function defined by y = 250(1.7)*. The growth factor 1 + r is the base 1.7. To determine the growth rate, solve the equation 1 + r = 1.7 for r. r = 0.7 or 70% b. If the growth rate of a function is 5%, determine the growth factor. If r = 5% or 0.05, the growth factor is 1 + r = = a. Determine the growth factor in the Charlotte population function P = 541(1.032)'. b. Determine the growth rate. Express your answer as a percent. 6. a. Using the function defined by P = 541(1.032)', determine the population of Charlotte in That is, determine P when t 6. b. Graph the population function with a graphing calculator. Set the window to Xmin = -50, Xmax = 100, Ymin = 0, and Ymax = 13,000. The graph should appear as follows: c. Determine P when t = 0. What is the graphical and the practical meaning of this number?

4 584 CHAPTER 5 MODELING WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS 7. a. Use the population model to predict Charlotte's population in b. Verify your prediction on the graph. 8. a. Use the graph to estimate when Charlotte's population will reach 700,000, assuming it continues to grow at the same rate. Remember, P is the number of thousands. b. Evaluate P when r = 32 and describe what it means. 9. Use the population model to estimate the population of Charlotte in 2002 and in In which prediction are you more confident? Why? 10. a. Assuming the growth rate remains constant, how long will it take for Charlotte to double its 2000 population? b. Explain how you reached your conclusion in part a. Waste-Water Treatment Facility You are working at a waste-water treatment facility. You are presently treating water contaminated with 18 micrograms of pollutant per liter. Your process is designed to remove 20% of the pollutant during each treatment. Your goal is to reduce the pollutant to less than 3 micrograms per liter. 11. a. What percent of pollutant present at the start of a treatment remains at the end of the treatment? b. The concentration of pollutant is 18 micrograms per liter at the start of the first treatment. Use the result of part a to determine the concentration of. pollutant at the end of the first treatment.

5 : ACTIVITY 5.6 POPULATION GROWTH 585 c. Complete the following table. Round the results to the nearest hundredth. n, NUMBER OFTREATMENTS 0 1 ff 2Í t ^ M?» 5 C, CONCENTRATION OF POLLUTANT, IN ng/l, ATTHE END OF THE HTH TREATMENT d. Write an equation for the concentration, C, of the pollutant as a function of the number of treatments, n. The equation C = 18(0.80)" has the general form C = C 0 (l - r) n, where ris rm. the decay rate, (1 r) is the decay factor or the base of the exponential function, n is the number of treatments, and C 0 is the initial value, the concentration when n = 0. Example 5 a. Determine the decay factor and the decay rate of the function defined by y = 123(0.43)*. The decay factor 1 - r is the base, To determine the decay rate, solve the equation \ r 0.43 for r. r = 0.57 or 57% b. If the decay rate of a function is 5%, determine the decay factor. If r = 5% or 0.05, the decay factor is 1 - r = = a. If the decay rate is 2.5%, what is the decay factor? b. If the decay factor is 0.76, what is the decay rate? 13. a. Use the function defined by C = 18(0.8)" to predict the concentration of contaminants at the waste-water treatment facility after seven treatments. b. Sketch a graph of the concentration function on a graphing calculator. Use the table in Problem lie to set a window. Does the graph look like you expected it would? Explain.