Question

the current fox population in a protected region is 367 and is decreasing by 8% 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 9 years from now. (1 point) \bigcirc $f(x) = 367 \cdot 0.92^x$ the estimated fox population 9 years from now will be 173. \bigcirc $f(x) = 367 \cdot 0.98^x$ the estimated fox population 9 years from now will be 306 \bigcirc $f(x) = 367 \cdot 1.08^x$ the estimated fox population 9 years from now will be 734. \bigcirc $f(x) = 367 \cdot 0.92x$ the estimated fox population 9 years from now will be 3,039.

Explanation:

Step1: Determine the exponential decay formula

For exponential decay, the formula is \( f(x)=a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decrease, and \( x \) is time. Here, \( a = 367 \), \( r=0.08 \), so \( 1 - r=1 - 0.08 = 0.92 \). Thus, the function is \( f(x)=367\cdot0.92^{x} \).

Step2: Calculate population at \( x = 9 \)

Substitute \( x = 9 \) into the function: \( f(9)=367\cdot0.92^{9} \). Calculate \( 0.92^{9}\approx0.471 \). Then \( 367\times0.471\approx367\times0.471\approx173 \).

Answer:

\( f(x) = 367\cdot0.92^{x} \) The estimated fox population 9 years from now will be 173.