The current fox population in a protected region is 367 and is decreasing by 8% per year. Find the exponential function

Question

that represents the fox population, where
is the number of years from now. Estimate the fox population of the region 9 years from now.

A
f(x)=367⋅0.92xf\left(x\right)=367\cdot0.92^xf(x)=367⋅0.92
x
. The estimated fox population 9 years from now will be 173

B
f(x)=367⋅1.08xf\left(x\right)=367\cdot1.08^xf(x)=367⋅1.08
x
. The estimated fox population 9 years from now will be 734

C
f(x)=367⋅0.98xf\left(x\right)=367\cdot0.98^xf(x)=367⋅0.98
x
. The estimated fox population 9 years from now will be 306

D
f(x)=367⋅0.92xf\left(x\right)=367\cdot0.92^xf(x)=367⋅0.92
x
. The estimated fox population 9 years from now will be 3,039

Answer 1

To find the exponential function that represents the fox population, we need to account for the 8% decrease per year. If the current population is \( P_0 = 367 \), the population after \( x \) years can be modeled as:

\[ f(x) = P_0 \cdot (1 - r)^x \]

where \( r \) is the rate of decrease (8% or 0.08). Therefore, the function becomes:

\[ f(x) = 367 \cdot (1 - 0.08)^x = 367 \cdot (0.92)^x \]

So, the correct form of the exponential function is:

\[ f(x) = 367 \cdot 0.92^x \]

Now, we can calculate the estimated fox population 9 years from now by plugging \( x = 9 \) into the function:

\[ f(9) = 367 \cdot 0.92^9 \]

Calculating \( 0.92^9 \):

\[ 0.92^9 \approx 0.422 \]

Now multiply by 367:

\[ f(9) \approx 367 \cdot 0.422 \approx 155.59 \approx 156 \text{ (when rounded)} \]

None of the provided options gives this result for the estimated population after 9 years. However, let's check the options again:

A: \( 173 \)
B: \( 734 \)
C: \( 306 \)
D: \( 3,039 \)

Based on the calculations, it appears the closest option to the calculated population would be A, but it states \( 173 \) instead of the \( 156 \) we found.

Conclusion

The correct exponential function is \( f(x) = 367 \cdot 0.92^x \), but the estimated population after 9 years should be approximately \( 156 \), which doesn't exactly match any of the given answers. It appears there might be an error in the choices provided.