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?

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21-10-2014
Mathematics

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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?

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teddydebevoise teddydebevoise · Answer 1 · 2014-10-21T02:15:47+02:00

The maximum occurs when the derivative of the function is equal to zero.
[tex]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[/tex]
Then evaluate the function for that time to find the maximum population.
[tex]P(t)=-0.3t^{4}+130t \\ P(4.76702)=-0.3*4.76702^{4}+130*4.76702+1100 \\ P(4.76702)=1564.79201[/tex]
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.