The current fox population in a protected region is 640 and is decreasing by 3% per year. Find the exponential function f(x)

Question

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 start with the initial population of 640 foxes, which is decreasing by 3% per year. This means that each year, the population is 97% (or 0.97) of the previous year's population.

The exponential function representing the fox population, where \( x \) is the number of years from now, is given by:

\[ f(x) = 640(0.97)^x \]

Now, we need to estimate the fox population 10 years from now. We can do this by substituting \(x = 10\) into the function:

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

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

\[ (0.97)^{10} \approx 0.7374 \]

Now, substituting this value back into the population function:

\[ f(10) \approx 640 \times 0.7374 \approx 471.936 \]

Rounding this value, we can estimate that the fox population 10 years from now will be approximately 472.

Thus, the correct choice is:

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