Question

the fox population in a certain region has a continuous growth rate of 6 percent per year. it is estimated that the population in the year 2000 was 15000.
(a) find a function that models the population t years after 2000 (t = 0 for 2000).
hint: use an exponential function with base e.
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)

Explanation:

Step1: Determine the general form of the exponential function

The general form of an exponential function 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. Given \(P_0 = 15000\) and \(r=0.06\), the function is \(P(t)=15000e^{0.06t}\).

Step2: Calculate the population in 2008

For the year 2008, \(t = 2008 - 2000=8\). Substitute \(t = 8\) into the function \(P(t)=15000e^{0.06t}\).

$$
LATEXBLOCK0
$$

Rounding to an integer, we get \(24243\).

Answer:

(a) \(P(t)=15000e^{0.06t}\)
(b) \(24243\)