Ecologists model the population density of a species in a circular wildlife reserve…

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Ecologists model the population density of a species in a circular wildlife reserve by

p(A)=3A+\dfrac{200}{A}

animals per square kilometer, where

A

is the area of the reserve in square kilometers. The area depends on the radius

r

(in kilometers) through

A=\pi r^{2}

. If the radius is expanding at

0.04\text{ km/yr}

when

r=5\text{ km}

, which of the following is the best approximation of

\dfrac{dp}{dt}

at that instant? (a)

12.6\;\text{animals}/\,\text{km}^{2}\,\text{per yr}

(b)

0.13\;\text{animals}/\,\text{km}^{2}\,\text{per yr}

(c)

3.7\;\text{animals}/\,\text{km}^{2}\,\text{per yr}

(d)

-3.7\;\text{animals}/\,\text{km}^{2}\,\text{per yr}