The rabbit population on a small island is observed to be given by the function P(t) = 130t − 0.3t^4 + 1100 where t is the time since observations of the island began. When is the maximum population attained What is the maximum population? When does the rabbit population disappear from the island?

The maximum occurs when the derivative of the function is equal to zero.

\( P(t)=-0.3t^{4}+130t+1100 \\ P’(t)=-1.2t^{3}+130 \\ 0=-1.2t^{3}+130 \\ 1.2t^{3}=130 \\ t^{3}= \frac{325}{3} \\ t=4.76702 \)

Then evaluate the function for that time to find the maximum population.

\( P(t)=-0.3t^{4}+130t \\ P(4.76702)=-0.3*4.76702^{4}+130*4.76702+1100 \\ P(4.76702)=1564.79201 \)

Depending on the teacher, the "correct" answer will either be the exact decimal answer or the greatest integer of that value since you cannot have part of a rabbit.

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