Question

6.7: homework assignment
score: 9.5/10 answered: 9/10
question 10
the fox population in a certain region has a continuous growth rate of 5 percent per year. it is estimated that the population in the year 2000 was 10600.
(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 with 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 \(P_0 = 10600\) and \(r=0.05\), so \(P(t)=10600e^{0.05t}\).

Step2: Calculate the population in 2008

For the year 2008, \(t = 2008 - 2000=8\).
Substitute \(t = 8\) into the function \(P(t)=10600e^{0.05t}\), we get \(P(8)=10600e^{0.05\times8}\).
First calculate \(0.05\times8 = 0.4\), then \(e^{0.4}\approx1.49182\).
So \(P(8)=10600\times1.49182\approx15813.292\).
Since the answer must be an integer, \(P(8)\approx15813\).

Answer:

(a) \(P(t)=10600e^{0.05t}\)
(b) \(15813\)