Expert: Paul Klarreich - 12/10/2011

Question
Hello, I've looked through your previously worked related rates problem, but I'm still having a little trouble. The question is as follows:

Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeters, the radius is increasing at the rate of 2.5 centimeter per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time in centimeters/minute?

I understand that Volume of right circular cylinder is =pi(radius)^2(height), therefore I'm assuming the volume of the cylinder is 5000*pi. I've also determined that (dr/dt)= 2.5 cm/min and that I'm solving for (dh/dt). My question concerns writing an equation with the two different variable so that I may substitute in (dr/dt) and (dh/dt) to find the final rate of change for the height of the oil slick.

Thank you very much :)

Answer
Questioner: Renee
Country: Indiana, United States
Category: Calculus
Private: No
Subject: Related Rates
Question: Hello, I've looked through your previously worked related rates problem, but I'm still having a little trouble. The question is as follows:

Oil is leaking from a pipeline on the surface of a lake and forms an oil slick

>>>>>>>>>>>>>>  Ah yes, the infamous BP Problem.

whose volume increases at a constant rate of 2000 cubic centimeters per minute.

>>>>>>>>>>>>>>  So dV/dt = 2000

The oil slick takes the form of a right circular cylinder with both its radius and height changing with time.

..............................
r and h are variables,

V = pi r^2 h

Differentiate implicitly w.r.t. time:

dV/dt = pi( 2rh dr/dt + r^2 dh/dt)   <<< product rule used here.
...............................

At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeters, the radius is increasing at the rate of 2.5 centimeter per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time in centimeters/minute?

This says:

When  r = 100,  h = 0.5,   dr/dt = 2.5,  now find  dh/dt

Substitute:  

dV/dt = pi( 2rh dr/dt + r^2 dh/dt)

2000  = pi( 2rh dr/dt + r^2 dh/dt)   <<<< use dV/dt = 2000 from before

2000  = pi( 2(100)h dr/dt + (100)^2 dh/dt)   <<<< r = 100

2000  = pi( 2(100)(0.5) dr/dt + (100)^2 dh/dt)  <<<< h = 0.5

2000  = pi( 2(100)(0.5)(2.5) + (100)^2 dh/dt)   <<<<  dr/dt = 0.5

Substitutions are complete, now solve for  dh/dt.