Answer:
Part A)
About 0.51% per year.
Part B)
About 0.30% per year.
Part C)
About 28.26%.
Step-by-step 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:
[tex]f(t) = 10.72(0.9t+10)^{0.3}\text{ where } 0 \leq t \leq 20[/tex]
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:
[tex]\displaystyle f'(t) = \frac{d}{dt} \left[ 10.72\left(0.9t+10\right)^{0.3}\right][/tex]
Rewrite:
[tex]\displaystyle f'(t) = 10.72\frac{d}{dt} \left[(0.9t+10)^{0.3}\right][/tex]
We can use the chain rule. Recall that:
[tex]\displaystyle \frac{d}{dx} [u(v(x))] = u'(v(x)) \cdot v'(x)[/tex]
Let:
[tex]\displaystyle u(t) = t^{0.3}\text{ and } v(t) = 0.9t+10 \text{ (so } u(v(t)) = (0.9t+10)^{0.3}\text{)}[/tex]
Then from the Power Rule:
[tex]\displaystyle u'(t) = 0.3t^{-0.7}\text{ and } v'(t) = 0.9[/tex]
Thus:
[tex]\displaystyle \frac{d}{dt}\left[(0.9t+10)^{0.3}\right]= 0.3(0.9t+10)^{-0.7}\cdot 0.9[/tex]
Substitute:
[tex]\displaystyle f'(t) = 10.72\left( 0.3(0.9t+10)^{-0.7}\cdot 0.9 \right)[/tex]
And simplify:
[tex]\displaystyle f'(t) = 2.8944(0.9t+10)^{-0.7}[/tex]
For 2002, t = 2. Then the rate at which the percentage is changing will be:
[tex]\displaystyle f'(2) = 2.8944(0.9(2)+10)^{-0.7} = 0.5143...\approx 0.51[/tex]
Contextually, this means the percentage is increasing by about 0.51% per year.
Part B)
Evaluate f'(t) when t = 17. This yields:
[tex]\displaystyle f'(17) = 2.8944(0.9(17)+10)^{-0.7} =0.3015...\approx 0.30[/tex]
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:
[tex]\displaystyle f(17) = 10.72(0.9(17)+10)^{0.3}=28.2573...\approx 28.26[/tex]
So, about 28.26% of the American population in 2017 are age 55 and older.
