Question

current learning objective: investigating continuous growth
question 8
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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 16300.
(a) find a function that models the population t years after 2000 (t = 0 for 2000).
p(t) = enter your answer
(b) use the function from part (a) to estimate the fox population in the year 2008.
your answer is (the answer must be rounded to the nearest whole number)
population = enter your answer
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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 \(P_0 = 16300\) and \(r=0.05\) (since \(5\%=0.05\)). So the function is \(P(t)=16300e^{0.05t}\).

Step2: Calculate the population in 2008

For the year 2008, \(t = 2008 - 2000=8\). Substitute \(t = 8\) into the function \(P(t)=16300e^{0.05t}\). Then \(P(8)=16300e^{0.05\times8}=16300e^{0.4}\). Using a calculator, \(e^{0.4}\approx1.49182\). So \(P(8)=16300\times1.49182\approx24317\).

Answer:

(a) \(P(t)=16300e^{0.05t}\)
(b) \(24317\)