To model the population using an exponential growth function of the form $P(t) = P_0 \cdot 2^{\frac{t}{n}}$, where:

• $P(t)$ is the population at time $t$ years,
• $P_0$ is the initial population,
• $n$ is the number of years it takes for the population to double,
• $t$ is the time in years from the start.

From the problem, we know the population doubled over a 6-year period, and the current population is 31,647.

Given that the population doubled in 6 years, $n = 6$. Therefore, the exponential function becomes:

$P(t) = P_0 \cdot 2^{\frac{t}{6}}$

However, we need to find $P_0$, the population at $t = 0$, i.e., 6 years ago, because the current population $P(6)$ is 31,647.

Since the population doubled in 6 years: $P(6) = 2 \cdot P_0 = 31,647$ $P_0 = \frac{31,647}{2} = 15,823.5$

Thus, the exponential function that models the population is:

$P(t) = 15,823.5 \cdot 2^{\frac{t}{6}}$

Would you like further clarification or any more details?

Here are some related questions:

1. How do we verify the model with other population data points?
2. What happens to the population over 12 or 18 years?
3. How does changing the value of $n$ affect the function?
4. Can you model population decline with a similar exponential function?
5. How can you derive a similar equation for other growth rates?

Tip: For exponential growth, knowing how long it takes for the quantity to double (doubling time) makes modeling easier and faster.