This problem involves exponential population growth, where the population grows at a rate proportional to the number of people present. The mathematical model for exponential growth is:

$P(t) = P_0 e^{kt}$

Where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population (at $t = 0$),
• $k$ is the growth rate constant,
• $t$ is the time in years.

1. Finding the initial population $P_0$

We are given two pieces of information:

1. The population doubles in 5 years. This implies $P(5) = 2P_0$.
2. The population is 8,000 after 3 years, i.e., $P(3) = 8000$.

Let's first use the second piece of information to express the population after 3 years: $P(3) = P_0 e^{3k} = 8000$

Now, from the first piece of information, we know that the population doubles in 5 years, so: $P(5) = P_0 e^{5k} = 2P_0$

We can divide these two equations to eliminate $P_0$ and solve for $k$:

$\frac{P(5)}{P(3)} = \frac{P_0 e^{5k}}{P_0 e^{3k}} = \frac{2P_0}{8000}$

This simplifies to:

$e^{2k} = \frac{2P_0}{8000}$

Solving for