514 Chapter 3 Exponential and Logarithmic Functions 82. 60+ Years after Brown .

Question

514 Chapter 3 Exponential and Logarithmic Functions 82. 60+ Years after Brown . Board" In 1954, the Supreme Court ended legal segregation in LUS public schools. The bar graph shows the percentage of black and white U.S adults ages 25 or older who completed high school since the end of segregation. U.S. Adults Ages 25 or Older Who Completed High School 92 100%[ Blacks 86 83 ? Whites 80% 60% 49 20% 2012 1992 Year 1962 Source: Pew Rescarch Center The functions B(x) = 16 Inx + 23 and w(r)-11 Inr + 49 model the percentage of black adults, B(a), and white adults W(x), who had completed high school x years aftet 1961. a. Use the appropriate model to determine the percentage of white adults who had completed high school in 2012 Round to the nearest percent. How well does this rounded valuc model the percents displayed by the graph? b. Use the appropnate model to determine when 90%? black adults will have completed high school. Round to the nearest year Use the formula for compound interest with n compoundings per year to solve this problem. How long, to the nearest tenth of a year, will it take $12,500 to grow to S20000 at 65% 83. annual interest compounded quarterly? Use the formala for continuous compounding to solve Exercises 84-85 84. How long, to the nearest tenth of a year, will it take $50,000 to triple in value at 7.5% annual interest compounded continuously? 85. What interest rate, to the nearest percent, is required for an invesiment subject to continuous compounding to triple in

Explanation / Answer

(a) Percentage of white adults W(x) is given by the Model

W(x) = 11 ln x + 49. The number of years (x), elapsed since 1961 till 2012 is 51.

Therefore W(x)= 11 ln51 +49

                        =43.25 +49 = 92 (nearest percent)

Percentage displayed in the graph is 92. Hence both the figures are matching.

(b) Percentage of black adults B(x) is given by the model

B(x)= 16 ln x +23. Here B(x) is given to be 90, we are required to figure out x. Accordingly,

90 = 16 ln x +23

16ln x= 90-23=67

lnx= 67/16=4.1875

x= e4.1875 =65.8579=66 yrs (round figure)

Base year is 1961 hence the 90% target about black adults would be reached in the year 1961+66= 2027

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