The U.S. Bureau of the Census prediction for the percentage of the population 65

Question

The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modemed as p(x) = -0.00022x^3 + 0.14x62 - 0.0033x + 12.236 percent where x is the number of years since 2000, data from 0 lessthanorequalto x lessthanorequalto 50. Determine the value of x in the domain 0 lessthanorequalto x lessthanorequalto 50 for with the percentage is predicted to be increasing most rapidly Thus, In which year is the percentage is predicted to be increasing most rapidly? Calculate the percentage at that time. Calculate the rate of change of the percentage at that time.

Explanation / Answer

Given

p(x) = -0.00022*x^3 + 0.014*x^2 - 0.0033*x + 12.236

derivative of x

dp(x)/dx = -0.00066x^2 + 0.028X - 0.0033

equating to 0

0 = -0.00066x^2 + 0.028X - 0.0033

x = 0.118

x = 42.306

curve has positive slope all along the orginal curve

Look at the following excel sheet

a) the values are increasing maximum at points 21 and 22 for x ( determined by highest slope)

b) Thus in year 2021 the percentage is predicted to be increasing more rapidly to 2022

c) The p(x) value at the year 2021 or 2022 where slope is maximum is 16.5964

d) Rate of change in percentage between 2022 and 2021 is (16.5964-16.30328) = 0.29356

x p(x) slope = p(x2)-p(x1) 1 12.24648 2 12.28364 0.03716 3 12.34616 0.06252 4 12.43272 0.08656 5 12.542 0.10928 6 12.67268 0.13068 7 12.82344 0.15076 8 12.99296 0.16952 9 13.17992 0.18696 10 13.383 0.20308 11 13.60088 0.21788 12 13.83224 0.23136 13 14.07576 0.24352 14 14.33012 0.25436 15 14.594 0.26388 16 14.86608 0.27208 17 15.14504 0.27896 18 15.42956 0.28452 19 15.71832 0.28876 20 16.01 0.29168 21 16.30328 0.29328 22 16.59684 0.29356 23 16.88936 0.29252 24 17.17952 0.29016 25 17.466 0.28648 26 17.74748 0.28148 27 18.02264 0.27516 28 18.29016 0.26752 29 18.54872 0.25856 30 18.797 0.24828 31 19.03368 0.23668 32 19.25744 0.22376 33 19.46696 0.20952 34 19.66092 0.19396 35 19.838 0.17708 36 19.99688 0.15888 37 20.13624 0.13936 38 20.25476 0.11852 39 20.35112 0.09636 40 20.424 0.07288 41 20.47208 0.04808 42 20.49404 0.02196 43 20.48856 -0.00548 44 20.45432 -0.03424 45 20.39 -0.06432 46 20.29428 -0.09572 47 20.16584 -0.12844 48 20.00336 -0.16248 49 19.80552 -0.19784 50 19.571 -0.23452

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