Optimization and Calculus problem: Rate of change of oil slick?

Aug 2010
1
0
The more i read the question to this problem, the more i understand what it is asking. However, i cannot develop an equation to solve for the rate of change.

Q: A circular oil slick spreads on the surface of the ocean, the result of a spill of 10 ft^3 of oil. When its radius is 50 ft, its radius is increasing at the rate of 2 ft/s. How fast is the thickness of the slick changing then?

A: - 1 / ( 3125pi ) ft/s

I believe the volume of oil was originally 10 ft cubed. Then the spill creates a oil slick with a radius of 50ft and some "unknown" amount of depth. (oil has to have volume even in the ocean)
So as the oil begins to rise to the surface because it is less dense than water, which increases the radius of the current oil slick at a rate of 2 ft/s. Now the thickness of the oil slick must be decreasing, but at what rate? <---- this is how i interpreted the question above, i could be completely wrong.

Any help is appreciated.(Talking)
 
Jun 2009
806
275
Volume of oil V = π*r^2*x, where x is he thickness of the oil at any instant. Here V is constant. Only the radius and the thickness change with time.

So V/x = π*r^2.

Taking the derivative with respect to the time, we get

V(-1/x^2)*dx/dt = 2*π*r*dr/dt

dx/dt = -2*π*r*dr/dt*x^2/V = -2*π*r*dr/dt*(V^2/π^2*r^4)*(1/V)

Now simplify and find dx/dt