Population Change The rabbit population on a small island is...

Population Change The rabbit population on a small island 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 island
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

Roots of Functions

Roots, or zeros, of a function are the values for which the function equals zero. In the context of modeling, finding these roots can indicate significant events, such as the point in time when a population may reach zero, signaling disappearance or extinction.

Population Modeling

Population modeling applies mathematical functions to describe and predict the behavior of populations over time. It typically involves using functions to incorporate factors that influence growth and decline, and applying calculus to analyze trends like maximum population sizes or extinction points.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number exponents, used to model various real-world phenomena. They provide a way to represent complex relationships, such as population change, through a combination of linear and higher-order terms, where the degree of the polynomial influences the curvature and behavior of the graph.

Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function, which is often essential in determining key points such as peak population levels or resource usage. Utilizing techniques like taking derivatives to find critical points allows one to identify where the function attains its highest or lowest values.

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