Answer:
Explanation:
To predict the population of the small town in 2025 using an exponential equation, we can use the formula for exponential decay:
\[ P(t) = P_0 \times e^{kt}, \]
where:
- \( P(t) \) is the population at time \( t \) (in this case, 2025),
- \( P_0 \) is the initial population (in 2000),
- \( e \) is the base of the natural logarithm (approximately 2.71828),
- \( k \) is the decay constant (which is negative for decay),
- \( t \) is the time in years.
Given that the population in 2000 (\( P_0 \)) was 12,847 and the decay rate is 0.4%, we can calculate the decay constant (\( k \)) as follows:
\[ k = \frac{\ln(1 - \text{decay rate})}{-\text{years}} \]
Substituting the values:
\[ k = \frac{\ln(1 - 0.004)}{-25}. \]
Now, we can use this value of \( k \) in the exponential decay formula to predict the population in 2025 (\( P(2025) \)):
\[ P(2025) = 12847 \times e^{k \times 25}. \]
Let's calculate the value of \( P(2025) \):
\[ k = \frac{\ln(1 - 0.004)}{-25} \approx -0.01606. \]
\[ P(2025) = 12847 \times e^{-0.01606 \times 25} \approx 11532. \]
So, based on the exponential decay model, the predicted population of the small town in 2025 is approximately 11,532.