Population Change The rabbit population on a small is- land...

Population Change The rabbit population on a small is- land is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where $t$ is the time (in months) since observations of the island began. (a) When is the maximum population attained, and what is that maximum population? (b) When does the rabbit population disappear from the island?

Population Change The rabbit population on a small is- land is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where $t$ is the time (in months) since observations of the island began.
(a) When is the maximum population attained, and what is that maximum population?
(b) When does the rabbit population disappear from the island?

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Key Concepts

Critical Points and Derivatives

In calculus, finding the maximum or minimum of a function often involves calculating its derivative and solving for points where the derivative equals zero. These critical points indicate where the function’s rate of change transitions, and applying tests like the second derivative test can confirm if these points correspond to a maximum, minimum, or a point of inflection.

Solving Polynomial Equations

When a model, such as a population function, is represented as a polynomial, determining key events involves solving polynomial equations. This typically requires setting the function equal to a specific value (such as zero for extinction), and applying algebraic manipulation or numerical approximation methods to find the roots, which correspond to the timings of those events.

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