Chapter 4.3, Problem 56E

Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at $8$ A.M. Assume that the pollutant disperses horizontally, forming a circle. If $t$ represents the time (in hours) since the factory began emitting pollutants $(t=0$ represents 8 A.M.), assume that the radius of the circle of pollution is $r\left(t\right)=2t$ miles. Let $A\left(r\right)=\pi r^2$ represents the area of a circle of radius $r$.

a. Find and interpret $A\left[r\left(t\right)\right]$.

b. Find and interpret $D_{t}A\left[r\left(t\right)\right]$ when $t=4$.