The fox population in a certain region has an annual growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 19300.
(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 should be an integer)

Step 1 ：Translate the problem into mathematical terms. The problem is asking for a function that models the population growth of foxes in a certain region. The growth rate is 4 percent per year and the initial population in the year 2000 was 19300. This is a typical exponential growth problem. The general formula for exponential growth is: \(P(t) = P0 * e^{rt}\) where: \(P(t)\) is the population at time t, \(P0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time.

Step 2 ：Substitute the given values into the formula. In this case, \(P0\) is 19300, \(r\) is 0.04 (4 percent expressed as a decimal), and \(t\) is the number of years after 2000. So the function \(P(t) = 19300 * e^{0.04t}\) models the population t years after 2000.

Step 3 ：Use the function to estimate the fox population in the year 2008. To do this, substitute \(t = 8\) into the function to get \(P(8) = 19300 * e^{0.04*8}\).

Step 4 ：Simplify the expression to get the estimated fox population in the year 2008. The estimated fox population in the year 2008 is approximately 26579.

Step 5 ：Final Answer: (a) The function that models the population \(t\) years after 2000 is \(P(t) = 19300 * e^{0.04t}\). (b) The estimated fox population in the year 2008 is \(\boxed{26579}\).