​(a) If its population doubled in size over 6​-year period and the current population is 31,647, write an exponential function of the form P(t)=P_(0)*2^((1)/(n)) that models the population.
P(t)=

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.

If its population doubled in size over 6-year period and the current population is 31,647, write an exponential function of the form P(t)=P_(0)*2^((1)/(n)) that models the population.

Learn how to model population growth with an exponential function when the population doubles every 6 years. This problem involves calculating the initial population given a current population of 31,647, and using the formula P(t) = P_0 * 2^(t/6). Suitable for students in Grades 9-12.

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