Question

review for exam 3 over chapters 6 and 11
score: 19.25/26 answered: 22/26
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question 16
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 23900.
(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)
question help: video
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Explanation:

Step1: Recall the exponential 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 \(P_0 = 23900\) (population in 2000, when \(t = 0\)) and \(r=0.05\) (since \(5\%=0.05\)).
So, \(P(t)=23900e^{0.05t}\)

Step2: Calculate the population in 2008

For the year 2008, \(t = 2008 - 2000=8\)
Substitute \(t = 8\) into the function \(P(t)=23900e^{0.05t}\)
\(P(8)=23900e^{0.05\times8}\)
First, calculate \(0.05\times8 = 0.4\)
Then \(e^{0.4}\approx1.491825\)
\(P(8)=23900\times1.491825\)
\(P(8)=23900\times1.491825 = 35654.6175\approx35655\)

Answer:

(a) \(P(t)=23900e^{0.05t}\)
(b) \(35655\)