Answer
a) $P(t)=2.58+0.09 t$
The slope of the line tells us the the population of the country is growing at a rate of $0.09$ million people per year, or equivalently, by $90,000$ people per year.
b) $P(t)= 2.68\cdot\left(1.026\right)^t$
This model tells us the the population is growing at a rate of about $2.6\%$ per year.
Work Step by Step
a) Let $P(t)=b+m t$, where $m$, the slope, is given by:
$$
m=\frac{\Delta P}{\Delta t}=\frac{P(13)-P(7)}{13-7}=\frac{3.75-3.21}{13-7}=\frac{0.54}{6}=0.09 .
$$ and $$
\begin{aligned}
& 3.21=b+0.09(7) \\
& 3.21=b+0.63 \\
& 2.58=b
\end{aligned}
$$ Hence $$P(t)=2.58+0.09 t$$ The slope of the line tells us the the population of the country is growing at a rate of $0.09$ million people per year, or equivalently, by $90,000$ people per year.
b) Let $P(t)=a b^t$. We know that $P(7)=a b^7=3.21$ and $P(13)=a b^{13}=3.75$. So
$$
\begin{aligned}
\frac{P(13)}{P(7)} & =\frac{a b^{13}}{a b^7}=\frac{3.75}{3.21} \\
b^6 & =\frac{3.75}{3.21} \\
\left(b^6\right)^{1 / 6} & =\left(\frac{3.75}{3.21}\right)^{1 / 6} \\
b & =1.026
\end{aligned}
$$ and $$
\begin{aligned}
P(7) & =a(1.026)^7=3.21 \\
a & =\frac{3.21}{(1.026)^7}=2.68
\end{aligned}
$$ Hence $$P(t)= 2.68\cdot\left(1.026\right)^t$$ This model tells us the the population is growing at a rate of about $2.6\%$ per year.
