Question

a model for a countrys population is given by the function of t, where t is the number of years since 1960. find and graph the percentage rate of change of f(t) for 0 ≤ t ≤ 50. f(t)=0.32t + 18.1
year country 1 (millions) country 2 (millions)
1960 18 38
1970 21 53
1980 25 67
1990 28 81
2000 31 95
2010 34 110
the percentage rate of change is p(t)=□.

Explanation:

Step1: Recall the formula for percentage - rate of change

The formula for the percentage - rate of change of a function $y = f(t)$ is $p(t)=\frac{f^{\prime}(t)}{f(t)}\times100\%$. First, find the derivative of $f(t)$.

Step2: Differentiate $f(t)$

Given $f(t)=0.32t + 18.1$. Using the power - rule $\frac{d}{dt}(at + b)=a$ (where $a = 0.32$ and $b = 18.1$), we have $f^{\prime}(t)=0.32$.

Step3: Calculate $p(t)$

Substitute $f^{\prime}(t)$ and $f(t)$ into the percentage - rate of change formula:

$$p(t)=\frac{0.32}{0.32t + 18.1}\times100=\frac{32}{0.32t + 18.1}$$

Answer:

$\frac{32}{0.32t + 18.1}$