A wildlife reserve is maintained as a circular region whose radius in kilometers…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A wildlife reserve is maintained as a circular region whose radius in kilometers

r

varies with time

t

(in hours) according to

r(t)=2+0.5\sqrt{t}, \qquad t\ge 0.

The animal population density

\rho

(in animals per square kilometer) at a distance corresponding to radius

r

is modeled by

\rho(r)=\frac{1200}{1+0.1r^{2}}.

Let

A

denote the area of the reserve and

P

denote the total animal population, where

P(t)=\rho(r(t))\,A(r(t))

. Assume all functions are differentiable on their domains. (a) Write

A

as a function of

r

and compute

\dfrac{dA}{dr}

. Include units. (b) Compute

\dfrac{d\rho}{dt}

t=9

hours. Show all steps, and include units. (c) Using your results from parts (a) and (b), determine

\dfrac{dP}{dt}

t=9

hours. Express the units of your answer clearly. (d) Interpret the sign of

\dfrac{dP}{dt}

you found in part (c) in the context of the wildlife reserve’s population management.