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Question

The aged dependency ratio is defined as the number of individuals age 6565 or older per 100100 individuals ages 206420-64. The aging of the baby boomer generation along with medical advancements and lifestyle changes for all individuals have caused this ratio to rise, shaping society's plans for the needs of a greater number of older individuals. By using Social Security Administration data for selected years from 19901990 and projected to 20452045, the aged dependency ratio can be modeled by the function

A(t)=0.000497t3+0.0449t20.669t+22.3A(t)=-0.000497 t^3+0.0449 t^2-0.669 t+22.3

where tt is the number of years past 19901990.

(a) Find the critical points for this model, and classify them as relative maxima or relative minima.

(b) Interpret the points found in part (a)

Solution

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(a). To find the critical points, we need to find the derivative of the given function and equate it to zero.

A(t)=0.000497t3+0.0449t20.669t+22.3A(t)=d(0.000497t3+0.0449t20.669t+22.3)dt=1491t21000000+449t50006691000=1491t289800t+6690001000000\begin{aligned} A(t)&=-0.000497t^3+0.0449t^2-0.669t+22.3\\ A'(t)&=\frac{d(-0.000497t^3+0.0449t^2-0.669t+22.3)}{dt}\\ &=-\frac{1491t^2}{1000000}+\frac{449t}{5000}-\frac{669}{1000}\\ &=-\frac{1491t^2-89800t+669000}{1000000} \end{aligned}

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