A wildlife preserve expands its fenced habitat at a time–dependent rate
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A wildlife preserve expands its fenced habitat at a time–dependent rate. The total enclosed area (in square kilometers) satisfies A(t)=π(5+0.6t)2, where t is measured in years. The density of a certain animal species within the habitat, in animals per square kilometer, is modeled by D(A)=80e−0.02A.
Both A and D are differentiable.
(a) Determine dtdD at t=4 years. Show all work and state units.
(b) Explain whether dtdD is expected to remain negative for all future times under this model, and justify your conclusion using properties of the given functions.