To determine the exponential function \( f(x) \) that represents the fox population, we start with the current population and the rate of decrease.
- The current population of foxes is \( 367 \).
- The decrease rate is \( 8% \) per year, which corresponds to retaining \( 100% - 8% = 92% \) of the population each year, or \( 0.92 \) of the population.
Therefore, the exponential function can be represented as follows:
\[ f(x) = 367 \cdot 0.92^x \]
Next, we want to estimate the fox population 9 years from now by calculating \( f(9) \):
\[ f(9) = 367 \cdot 0.92^9 \]
Calculating \( 0.92^9 \):
\[ 0.92^9 \approx 0.4225 \quad \text{(approximately)} \]
Now, we calculate \( f(9) \):
\[ f(9) \approx 367 \cdot 0.4225 \approx 155.43 \]
Rounding this result gives us approximately \( 155 \) foxes.
However, none of the provided responses include this estimate. Let's double-check the exponential function for accuracy before concluding:
- If the function was \( 367 \cdot 0.98^x \), that would indicate a population that increases, but we know the population is decreasing.
- If the function was \( 367 \cdot 1.08^x \), that would again indicate growth.
Only the option specifying \( f(x) = 367 \cdot 0.92^x \) makes sense for a decreasing population. However, the estimation from the provided options appears incorrect.
Thus, from the provided options, the correct function representation is:
\[ f(x) = 367 \cdot 0.92^x \]
Yet the estimated population 9 years from now is not among those provided answers. Please check the options or the calculations for discrepancies.
1 answer