The problem states that the current fox population is 367 and it is decreasing by 8% per year. To find the exponential function that represents the fox population, we can use the formula for exponential decay:
\[ f(x) = P \cdot (1 - r)^x \]
where:
- \( P \) is the initial population,
- \( r \) is the rate of decrease (in decimal form), and
- \( x \) is the number of years.
In this case:
- \( P = 367 \)
- \( r = 0.08 \) (which is 8%)
So, the function becomes:
\[ f(x) = 367 \cdot (1 - 0.08)^x = 367 \cdot 0.92^x \]
Now, to estimate the fox population 9 years from now, we can substitute \( x = 9 \) into the function:
\[ f(9) = 367 \cdot 0.92^9 \]
Calculating \( 0.92^9 \):
\[ 0.92^9 \approx 0.4224 \]
Now, we can calculate \( f(9) \):
\[ f(9) \approx 367 \cdot 0.4224 \approx 155.1 \]
So, the estimated fox population 9 years from now is approximately 155.
Now let's analyze the options provided in the question:
A: \( f(x) = 367 \cdot 0.92^x \). The estimated fox population 9 years from now will be 173.
B: \( f(x) = 367 \cdot 1.08^x \). The estimated fox population 9 years from now will be 734.
C: \( f(x) = 367 \cdot 0.98^x \). The estimated fox population 9 years from now will be 306.
D: \( f(x) = 367 \cdot 0.92^x \). The estimated fox population 9 years from now will be 3,039.
The correct exponential function is given by option A, but the estimated population of 173 in option A is incorrect based on our calculation. However, option D also uses the correct function but the population estimate is far too high.
Therefore, none of the options provided perfectly match the calculations. If the population decrease rate and calculated population is accurately considered, the proper expected population after 9 years should be roughly around 155, indicating a potential error in the answer choices provided.
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