The projected number of Americans 100 years of age and older for the years 2010
ID: 3000450 • Letter: T
Question
The projected number of Americans 100 years of age and older for the years 2010 to 2050 can be modeled by the functionC(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
(a) For 2040, x = 40
C(x = 40) = 0.017*40^3 - 1.16*40^2 + 30.83*40 -139.63
= 325.57
(b) For 2040, x = 40
C'(x) = 0.051x^2 - 2.32x + 30.83
C'(x = 40) = 0.051*40^2 - 2.32*40 + 30.83
= 19.63
(c) Since C'(x) represnts the growth rate of the population with passing years, it represents whether the population is increasing or decreasing. We got C'(x = 40) = 19.63. It means that the population is increasing at the rate of 19.63 persons per year in the year 2040
(d) % rate of change = rate of change/population *100
= (19.63/325.57) * 100
= 6.029 %
(e) It means that in 2040, the population is increasing at 6.029% per year
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