A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding at 0.1 m/min, and its thickness is .02 m. At that moment:

Question

a. How fast is the area of the slick expanding?
b. If the circular slick has the same thickness as everywhere, and the volume of oil spilled remains fixed, how fast is the thickness of the slick decreasing?

Answer 1

Pretty similar to the truck and the police car:

Area = A = pi r^2
dA/dt = 2 pi r dr/dt (which by the way is the circumference times the outward speed logically enough.)

part 2 is rate of volume change
Vol = V = pi r^2 h where h is thickness
dV/dt = pi r^2 (dh/dt) + h (2 pi r dr/dt)
but
dV/dt = 0 given so
pi r^2 dh/dt = - 2 pi r h dr/dt
we did dr/dt in part 1
dh/dt = - 2(h/r)dr/dt

Answer 2

-2.667* 10^-5