Question

the fox population in a certain region has a continuous growth rate of 7 percent per year. it is estimated that the population in the year 2000 was 7800.
(a) find a function that models the population t years after 2000 (t = 0 for 2000).
your answer is p(t) =
(b) use the function from part (a) to estimate the fox population in the year 2008.
your answer is (the answer must be an integer)
question help: video

Explanation:

Step1: Recall the continuous - growth formula

The formula for continuous growth is \(P(t)=P_0e^{rt}\), where \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years. Given \(P_0 = 7800\) and \(r=0.07\).
So, \(P(t)=7800e^{0.07t}\).

Step2: Calculate the population in 2008

For the year 2008, \(t = 2008 - 2000=8\).
Substitute \(t = 8\) into the function \(P(t)=7800e^{0.07t}\).
\(P(8)=7800e^{0.07\times8}=7800e^{0.56}\).
Using a calculator, \(e^{0.56}\approx1.75067\).
Then \(P(8)=7800\times1.75067 = 13655.226\approx13655\).

Answer:

(a) \(P(t)=7800e^{0.07t}\)
(b) \(13655\)