1. HOw FAST IS THE AREA OF THE SLICK EXPANDING?

Question

A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meter, the radius of the slick is expanding by 0.1 meter/min. and its thickness is 0.02 meter. At that moment:

1. How fast is the area of the slick expanding?

2. The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing?

Please answers with explanation

Answer 1

You've got all you need here to answer this one. For 1) take the differential equation for area of a circle dA=2*Pi*r*dr (you arive at this by integrating from a differential area in cylindrical coordinates, dA=dtheta*r*dr) put this with respect to time (basically divide both sides by dt) for dA/dt=2*Pi*r*dr/dt. Now plug in the appropriate values from the problem to solve for dA.

for part 2) approach this as with 1) but by solving for volume in cylindrical coordinates (dV=dtheta*dz*r*dr) until the change in volume is in terms of the knowns and the rate of change z (thickness). Put it with respect to time, and plug in the values given to solve for dz/dt (dV/dt is given, so the solution is algebraic)