Question

The current fox population in a protected region is 640 and is decreasing by 3% per year. Find the exponential function f(x)
that represents the fox population, where x
is the number of years from now. Estimate the fox population of the region 10 years from now.

A
f(x)=640(1.03)xf\left(x\right)=640\left(1.03\right)^xf(x)=640(1.03)
x
and the estimated fox population 10 years from now will be 860.

B
f(x)=640(0.97)xf\left(x\right)=640\left(0.97\right)^xf(x)=640(0.97)
x
and the estimated fox population 10 years from now will be 472.

C
f(x)=640(1.03)xf\left(x\right)=640\left(1.03\right)^xf(x)=640(1.03)
x
and the estimated fox population 10 years from now will be 472.

D
f(x)=640(0.97)xf\left(x\right)=640\left(0.97\right)^xf(x)=640(0.97)
x
and the estimated fox population 10 years from now will be 300.

Answer 1

To model the decreasing fox population, we need to use the exponential decay formula. The general formula for exponential decay is:

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

where:
- \( f(x) \) is the population at time \( x \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decrease (as a decimal), and
- \( x \) is the number of years from now.

In this case:
- The initial population \( P_0 = 640 \),
- The rate of decrease \( r = 0.03 \) (which corresponds to 3%).

Thus, we can plug these values into the formula:

\[
f(x) = 640 \cdot (1 - 0.03)^x = 640 \cdot (0.97)^x
\]

Now we will estimate the fox population 10 years from now:

\[
f(10) = 640 \cdot (0.97)^{10}
\]

Calculating \( (0.97)^{10} \):

\[
(0.97)^{10} \approx 0.737
\]

Now, substituting that back into the equation:

\[
f(10) \approx 640 \cdot 0.737 \approx 471.68
\]

Rounding this to the nearest whole number gives:

\[
f(10) \approx 472
\]

So the final function and estimation is:

The correct option is **B**:
\[
f(x) = 640(0.97)^x
\]
and the estimated fox population 10 years from now will be approximately 472.