To represent the fox population mathematically, we need to formulate an exponential decay function because the population is decreasing by a percentage each year.
The general formula for exponential decay can be written as: \[ f(x) = P_0 \cdot (1 - r)^x \] where:
- \( P_0 \) is the initial population,
- \( r \) is the decay rate (in decimal form),
- \( x \) is the number of years,
- \( (1 - r) \) represents the remaining population after each year.
In this case:
- The initial population \( P_0 = 367 \),
- The decay rate \( r = 0.08 \) (which corresponds to 8%).
Thus, \( (1 - r) = 1 - 0.08 = 0.92 \).
Now, we can write the function: \[ f(x) = 367 \cdot 0.92^x \]
Next, we need to estimate the fox population 9 years from now: \[ f(9) = 367 \cdot 0.92^9 \] Calculating \( 0.92^9 \): \[ 0.92^9 \approx 0.4224 \text{ (approximately)} \]
Now substituting this back into the function: \[ f(9) \approx 367 \cdot 0.4224 \approx 155.2 \text{ (approximately)} \]
Given the answer choices you provided, it appears none of the choices align accurately with our calculations, but you asked for the expected population. Based on rounding, you may want to consider \( 155 \) as the population estimate after 9 years, which isn't listed.
To summarize, the correct function representing the fox population is: \[ f(x) = 367 \cdot 0.92^x \] And the estimated fox population 9 years from now is approximately \( 155 \). None of the provided answer choices is correct based on this calculation.
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