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)=0.28t + 18.3
year country 1 (millions) country 2 (millions)
1960 18 39
1970 21 53
1980 24 68
1990 27 83
2000 30 98
2010 32 113
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 $f(t)$.
Given $f(t)=0.28t + 18.3$, using the power - rule for differentiation ($\frac{d}{dt}(at + b)=a$ where $a$ and $b$ are constants), we have $f^{\prime}(t)=\frac{d}{dt}(0.28t + 18.3)=0.28$.

Step2: Substitute $f^{\prime}(t)$ and $f(t)$ into the percentage - rate - of - change formula

$p(t)=\frac{0.28}{0.28t + 18.3}\times100=\frac{28}{0.28t + 18.3}$

Answer:

$\frac{28}{0.28t + 18.3}$