Conical Tank Water Leak Rate Calculation

[Feodalherren]

Homework Statement

Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3 / min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank.

Homework Equations

The Attempt at a Solution

This is how I started:

I want dV/dt when h=200 and dh/dt = 20.

I used similar triangles to get the radius of the smaller cone to be 1/√8

The volume of a cone is:
V=(1/3)∏hr^2

Last step was simply to differentiate the volume formula with the radius. Somewhere something went wrong, it just feels wrong to me... Help? :)

 

[Mark44]

Homework Statement

Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3 / min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank.

Homework Equations

The Attempt at a Solution

This is how I started:

I want dV/dt when h=200 and dh/dt = 20.

I used similar triangles to get the radius of the smaller cone to be 1/√8

This is where you went wrong. The radius of the smaller cone is changing all the time. Use similar triangles to get a relationship between the radius and height of the smaller cone. Then you can write the volume as a function of either h or r alone.

The volume of a cone is:
V=(1/3)∏hr^2

Last step was simply to differentiate the volume formula with the radius. Somewhere something went wrong, it just feels wrong to me... Help? :)

 

[Feodalherren]

Thank you I got it now! :)