The projected number of Americans 100 years of age and older for the years 2010

Question

The projected number of Americans 100 years of age and older for the years 2010 to 2050 can be modeled by the function
C(x)= 0.017x^3 - 1.16x^2 + 30.83x - 139.63

thousand people where x is the number of years after 2000.

Source of data: Population Division, U.S. Census Bureau (Based on Census 2000)
a. What is the projected number of Americans 100 years of age and older in 2040?
b. What is the derivative of this function in 2040?
c. Interpret the value you wrote in part b.
d. Find the percentage rate of change in the projected number in 2040.
e. Interpret the value you wrote in part d

Explanation / Answer

C(x)= 0.017x3 - 1.16x2 + 30.83x - 139.63

A) x=40

C(x=40)=0.017*(64000)-1.16*(1600)+30.83*40-139.63

            =1088-1856+1233.2-139.63=325.57

B,C) Derivative,dC/dx=3(0.017x2)-2(1.16x)+30.83

                                  =0.051(1600)-(2.32*40)+30.83

                                 =81.6-92.8+30.83=19.63

D,E)Percentage rate of change=(dC/C)*100

                                             =(19.63*100)/325.57

                                              =6.029%

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