The population density (in persons per square kilometer) of a circular city depends…

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

Problem

The population density

P

(in persons per square kilometer) of a circular city depends on its area

A

(in square kilometers) according to

P(A)=\frac{5000}{1+0.001A}.

The area satisfies

A=\pi r^{2}

, where the city’s radius

r

(in kilometers) changes with time following

r(t)=0.5t^{2}+2t

, with

t

in hours. Find

\dfrac{dP}{dt}

t=3

hours. Which value is closest to your result? (a)

-5.6\times10^{2}\;\text{persons/km}^{2}\!/\text{h}

(b)

9.1\times10^{2}\;\text{persons/km}^{2}\!/\text{h}

(c)

-9.1\times10^{2}\;\text{persons/km}^{2}\!/\text{h}

(d)

-1.2\times10^{3}\;\text{persons/km}^{2}\!/\text{h}