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5-Minute Check on Activity 5-5

5-Minute Check on Activity 5-5. Identify the y-intercept, growth or decay factor, and whether the function is increasing or decreasing y = (0.5)(1.25) x y = (6.2)(0.96) x What is constant with growth or decay problems?

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5-Minute Check on Activity 5-5

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  1. 5-Minute Check on Activity 5-5 • Identify the y-intercept, growth or decay factor, and whether the function is increasing or decreasing • y = (0.5)(1.25)x • y = (6.2)(0.96)x • What is constant with growth or decay problems? • If we have $100 that is growing at a rate of 10% per year, how long till it doubles? • If we have 10 grams of U235 that is decaying at a rate of 15% per month, how long till we have only 5 grams left? y-int = 0.5 GF = 1.25 increasing y-int = 6.2 DF = 0.96 decreasing the ratio between successive x-values Find intersection of y1 = 100(1.1)x and y2 = 200 x = 7.27 years Find intersection of y1 = 10(0.85)x and y2 = 10 x = 4.27 months Click the mouse button or press the Space Bar to display the answers.

  2. Activity 5 - 6 Population Growth Charlotte, NC 11/13/2005

  3. Objectives • Determine annual growth or decay rate of an exponential function represented by a table of values or an equation • Graph an exponential function having equation y = a(1 + r)x

  4. Vocabulary • None new

  5. Activity According to the 2000 US Census, the city of Charlotte, North Carolina, had a population of approximately 541,000. The population increases at a constant rate of 3.2%, determine the population of Charlotte (in thousands) in 2001. Determine the population of Charlotte (in thousands) in 2002. Divide the population in 2001 by the population in 2000 and record this ratio. 541,000 + 541,000 (.032) = 541,000 + 17,312 = 558,312 558,312 + 558,312 (.032) = 558,312 + 17,866 = 576,178 558,312  541,000 = 1.032

  6. Activity cont Divide the population in 2002 by the population in 2001 and record this ratio Are the ratios the same? What does this mean? Write an equation for Charlotte’s population: 576,178  558,312 = 1.032 Yes Ratios represent the growth factor P = 541,000(1.032)t

  7. Exponential Growth Rate r = b – 1 = 1.12 – 1 = .12 or 12% Determine the growth rate, r, given the following factors: • b = 1.12 • b = 1.07 • b = 1.33 Determine the growth factor, b, given the following rates: • r = 5.4% • r = 25% r = b – 1 = 1.07 – 1 = .07 or 7% r = b – 1 = 1.33 – 1 = .33 or 33% b = r + 1 = 0.054 + 1 = 1.054 b = r + 1 = 0.25 + 1 = 1.25

  8. Activity cont Fill in the table below for Charlotte’s population: What is the growth rate? What is the growth factor? What is the population of Charlotte in 2006? How long will it take for Charlotte’s population to double? r = 3.2% b = r + 1 = 0.032 + 1 = 1.032 P(6) = 541,000(1.032)6 ≈ 653,545 1082000 = 541,000(1.032)t Solve graphically t = 22.006 years

  9. Exponential Decay Rate r = 1 – b = 1 – 0.82 = .18 or 18% Determine the decay rate, r, given the following factors: • b = 0.82 • b = 0.87 • b = 0.93 Determine the decay factor, b, given the following rates: • r = 6.4% • r = 15% r = 1 – b = 1 – 0.87 = .13 or 13% r = 1 – b = 1 – 0.93 = .07 or 7% b = 1 – r = 1 – 0.064 = 0.936 b = 1 – r = 1 – .15 = 0.85

  10. Exponential Decay Example You are working at a waste-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. Complete the table: What percent of the pollutant remains after treatment? What is the concentration after the first treatment? b = 1 – r = 1 – 0.2 = 0.8 or 80% 18  0.8 = 14.4 micrograms

  11. Exponential Decay Example cont Write the equation for the concentration, C, of the pollutant as a function on the number of treatments, n. How many treatments are necessary to achieve the needed pollutant concentration? C = 18(0.8)n 3 = 18(0.8)n solve graphically n = 8.03 so after 9 treatments

  12. Summary and Homework • Summary • Exponential functions are used to describe phenomena that grow/decay by a constant percentage rate over time • Annual growth rate problems are modeled by P = P0(1 + r)twhere P0 is the initial amount, r is the annual growth rate, t is time in years, and (1 + r) represents the growth factor • Annual decay rate problems are modeled byP = P0(1 - r)twhere P0 is the initial amount, r is the annual decay rate, t is time in years and (1 - r) represents the decay factor • Homework • pg 586 – 588; problems 1, 2, 5

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