Question

The population of Americans age 55 and older as a percentage of the total population is approximated by the function f(t) = 10.72(0.9t + 10)^0.3 (0 <= t < = 20)

where t is measured in years, with t=0 corresponding to the year 2000.

Required:
a. At what rate was the percentage of Americans age 55 and older changing at the beginning of 2002?
b. At what rate will the percentage of Americans age 55 and older be changing in 2017?
c. What will be the percentage of the population of Americans age 55 and older in 2017?

Answer 1

Part A)

About 0.51% per year.

Part B)

About 0.30% per year.

Part C)

About 28.26%.

Explanation:

We are given that the population of Americans age 55 and older as a percentange of the total population is approximated by the function:

`f(t) = 10.72(0.9t+10)^(0.3)\text{ where } 0 \leq t \leq 20`

Where t is measured in years with t = 0 being the year 2000.

Part A)

Recall that the rate of change of a function at a point is given by its derivative. Thus, find the derivative of our function:

`\displaystyle f'(t) = (d)/(dt) \left[ 10.72\left(0.9t+10\right)^(0.3)\right]`

Rewrite:

`\displaystyle f'(t) = 10.72(d)/(dt) \left[(0.9t+10)^(0.3)\right]`

We can use the chain rule. Recall that:

`\displaystyle (d)/(dx) [u(v(x))] = u'(v(x)) \cdot v'(x)`

Let:

`\displaystyle u(t) = t^(0.3)\text{ and } v(t) = 0.9t+10 \text{ (so } u(v(t)) = (0.9t+10)^(0.3)\text{)}`

Then from the Power Rule:

`\displaystyle u'(t) = 0.3t^(-0.7)\text{ and } v'(t) = 0.9`

Thus:

`\displaystyle (d)/(dt)\left[(0.9t+10)^(0.3)\right]= 0.3(0.9t+10)^(-0.7)\cdot 0.9`

Substitute:

`\displaystyle f'(t) = 10.72\left( 0.3(0.9t+10)^(-0.7)\cdot 0.9 \right)`

And simplify:

`\displaystyle f'(t) = 2.8944(0.9t+10)^(-0.7)`

For 2002, t = 2. Then the rate at which the percentage is changing will be:

`\displaystyle f'(2) = 2.8944(0.9(2)+10)^(-0.7) = 0.5143...\approx 0.51`

Contextually, this means the percentage is increasing by about 0.51% per year.

Part B)

Evaluate f'(t) when t = 17. This yields:

`\displaystyle f'(17) = 2.8944(0.9(17)+10)^(-0.7) =0.3015...\approx 0.30`

Contextually, this means the percetange is increasing by about 0.30% per year.

Part C)

For this question, we will simply use the original function since it outputs the percentage of the American population 55 and older. Thus, evaluate f(t) when t = 17:

`\displaystyle f(17) = 10.72(0.9(17)+10)^(0.3)=28.2573...\approx 28.26`

So, about 28.26% of the American population in 2017 are age 55 and older.