Answer
a) $P=10(2.024)^t$
b) $t=6.53$ years
Work Step by Step
(a) Given that the population grows exponentially, it can be described by $P=a b^t$, where $P$ is the number of rabbits and $t$ is the number of years which have passed. The value of $a$ represents the initial number of rabbits, so $a=10$ and $P=10 b^t$. After 5 years, there are 340 rabbits so
$$
\begin{aligned}
340 & =10 b^5 \\
34 & =b^5 \\
\left(b^5\right)^{1 / 5} & =34^{1 / 5} \\
b & \approx 2.024 .
\end{aligned}
$$ Hence $P=10(2.024)^t$.
(b) We want to find $t$ when $P=1000$.
$$\begin{aligned}
10(2.024)^t&=1000\\
2.024^t&=100\\
t\ln 2.024&=\ln 100\\
t&=\frac{\ln 100}{\ln 2.024}\\
&\approx 6.53.
\end{aligned}$$ The population will reach $1000$ rabbits in around $t=6.53$ years.
