Question

the fox population in a certain region has a continuous growth rate of 9%. in the year 2012, there were 21100 foxes counted in the area. write a function, f(t), that models the number of foxes in the population at t years after 2012. f(t) = help (formulas) what is the fox population predicted to be in 2020? help (numbers) note: you can earn partial credit on this problem.

Explanation:

Step1: Determine the formula for continuous growth

The formula for continuous growth is \( f(t)=P_0e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years.
Given \( P_0 = 21100 \), \( r=0.09 \) (since \( 9\%=0.09 \)). So the function is \( f(t)=21100e^{0.09t} \).

Step2: Calculate the value of \( t \) for the year 2020

The year 2020 is \( 2020 - 2012=8 \) years after 2012. So \( t = 8 \).
Substitute \( t = 8 \) into the function: \( f(8)=21100e^{0.09\times8} \).
First calculate \( 0.09\times8 = 0.72 \). Then \( e^{0.72}\approx2.054433 \).
Multiply by 21100: \( f(8)=21100\times2.054433\approx43348.54 \).

Answer:

The function is \( f(t)=21100e^{0.09t} \) and the fox population in 2020 is approximately \( 43349 \).