Activity 3.6.4.
Suppose that a population of animals (measured in thousands) that lives on an island is known to grow according to the logistic model, where \(t\) is measured in years. We know the following information: \(P(0) = 2.45\text{,}\) \(P(3) = 4.52\text{,}\) and as \(t\) increases without bound, \(P(t)\) approaches \(11.7\text{.}\)
(a)
Determine the exact values of \(A\text{,}\) \(M\text{,}\) and \(k\) in the logistic model
\begin{equation*}
P(t) = \frac{A}{1 + Me^{-kt}}\text{.}
\end{equation*}
Clearly show your algebraic work and thinking.
(b)
Plot your model from (a) and check that its values match the desired characteristics. Then, compute the average rate of change of \(P\) on the intervals \([0,2]\text{,}\) \([2,4]\text{,}\) \([4,6]\text{,}\) and \([6,8]\text{.}\) What is the meaning (with units) of the values you’ve found? How is the population growing on these intervals?
(c)
Find the exact time value when the population will be \(10\) (thousand). Show your algebraic work and thinking.
