Related Rates Involving a Cone

[erok81]

Homework Statement

Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 5.00 inches at the top and a height of 7.00 inches. At the instant when the water in the container is 4.00 inches deep, the surface level is falling at a rate of 0.7 in/sec. Find the rate at which water is being drained from the container.

Homework Equations

v=1/3 pi r^2 h

The Attempt at a Solution

The values I came up with are as follows.

h=4 (of water level)
r=2.86 (of water level found using equal triangles)
dV/dt= what I am solving for
dh/dt = -0.7

Whenever I take the derivative of the volume equation I end up with a dr/dt that I have no idea what to do with. Am I just doing it wrong and getting the dr/dt when I shouldn't?

I also noticed in a few google results they'd used similar triangles to get r in terms of h. But whenever I do that, I get an actual value for r and the h goes away. Perhaps this is where I am messing up?

 

[erok81]

So I tried it without the r variable using r=5h/7 instead. That seemed to work because I got an answer that seems more reasonable, 17.955.

Is there any reason to do that? I am guessing because in this case you can only deal with two variables because of that dr/dt you end up with??

 

[HallsofIvy]

Yes, you can do that. The water takes the shape of the cone so no matter what the height of the water in the cone, you will always have r= (5/7) h. Draw a picture and think about "similar triangles".

 

[erok81]

So is the main reason why you use r= (5/7)h is to get rid of the r in the equation? Because I have the actual value for h. But if I plug it in, I end up with that dr/dh again that I don't know what to do wtih.

 

[Mark44]

Yes, you want to get rid of one variable so that you can write V as a function of a single variable, and then get dV/dh and dh/dt. After you have found those, then you can substitute the known values at the particular time.

 

[erok81]

Sweet. Thank you both. It is much appreciated.