Question

The current fox population in a protected region is 367 and is decreasing by 8 percent per year. Find the exponential function f(x) that represents the fox population, where x is the number of years from now. Estimate the population 9 years from now.

Answer 1

To find the exponential function \( f(x) \) that represents the fox population decreasing by 8 percent per year, we start from the initial population and apply the decay formula.

The general form for an exponential decay function is:

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

where:
- \( P_0 \) is the initial population,
- \( r \) is the decay rate (as a decimal),
- \( x \) is the number of years.

Given:
- \( P_0 = 367 \) (initial population),
- \( r = 0.08 \) (8 percent is converted to decimal).

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 will substitute \( x = 9 \) into the function:

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

First, we compute \( (0.92)^9 \):

\[
(0.92)^9 \approx 0.4224
\]

Now, multiply this value by the initial population:

\[
f(9) \approx 367 \cdot 0.4224 \approx 155.50
\]

Rounding to the nearest whole number, the estimated fox population 9 years from now is approximately \( 156 \).

Thus, the exponential function representing the fox population is:

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

And the estimated fox population 9 years from now is:

\[
\approx 156
\]