The population density (in hundred animals per square kilometer) of a protected…

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

Problem

The population density

D

(in hundred animals per square kilometer) of a protected habitat is given by

D(A)=0.5\sqrt{A}

, where

A

is the area of the habitat in square kilometers. The habitat is approximately circular with area

A=\pi R^{2}

, where

R

is its radius in kilometers. When

R=10\text{ km}

, the radius is increasing at a rate of

0.3\text{ km/day}

. What is

\dfrac{dD}{dt}

at that moment (in hundred animals per square kilometer per day)? (a)

2.66

(b)

0.27

(c)

-0.27

(d)

0.89