The population density (people per km ) inside a circular city is

MATHEMATICS · Subjective · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

The population density (people per km

^{2}

) inside a circular city is

D(A)=5000e^{-0.002A}

, where

A

is the city’s area in km

^{2}

. The area is related to the city’s radius by

A=\pi r^{2}

. When the radius is

10\text{ km}

and increasing at

\dfrac{dr}{dt}=0.30\text{ km/yr}

, determine

\dfrac{dD}{dt}

, the instantaneous rate of change of density with respect to time. Give a numerical answer rounded to the nearest whole person per km

^{2}

per year.