# Related Rates - Frustum of a Cone

1. Dec 6, 2013

### Temp0

1. The problem statement, all variables and given/known data

A large container has the shape of a frustum of a cone with top radius 9 metres , bottom radius 2 metres , and height 7 metres. The container is being filled with water at the constant rate of 4.2 cubic meters per minute.
At what rate is the level of water rising at the instant the water is 1 metre deep?

2. Relevant equations
V = 1/3∏ (r^2) * h
Internet Says the volume of a frustum of a cone is V = (∏ * h / 3) (R^2 + Rr + r^2)

3. The attempt at a solution
I don't know where to start with this one, I think I have to find some relationship between the height of the cone and the radius of the cone, however, I don't know which radius to use. Once I isolate h with V, I can just derive it and substitute 4.2 for dV/dt, 1 for h, and find dh/dt. Any tips on how to relate radius to height in this situation? Thanks in advance.

2. Dec 6, 2013

### vela

Staff Emeritus
Have you drawn a picture?

3. Dec 6, 2013

### HallsofIvy

Staff Emeritus
Looked at from the side, a frustrum of a cone is a trapezoid. If you set up a coordinate system with origin at the center of the smaller face, then the two sides are two lines, one passing through (r, 0) and (R, h). you can write the equation for that line and so get a relation between x (the radius) and y (the height) at each point.

4. Dec 6, 2013

### Temp0

Hmm, i've tried drawing a picture, but it didn't help.

That trapezoid idea might work, i'll try it, thanks!

5. Dec 6, 2013

### Staff: Mentor

A picture by itself probably won't help, but if you draw it on a coordinate system and identify the points on the picture by their coordinates, as HallsOfIvy suggests, it will be useful.

For problems like these it's always a good idea to draw a picture, as described above.

Last edited: Dec 6, 2013
6. Dec 6, 2013

### vela

Staff Emeritus
This is the picture I was wondering if you drew.

7. Dec 6, 2013

### SteamKing

Staff Emeritus
No picture.

8. Dec 6, 2013

### vela

Staff Emeritus
I meant I was wondering if he had drawn the trapezoid picture Halls had described. I didn't attach a picture.