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Water leaking out of a cone

by Feodalherren
Tags: cone, leaking, water
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Feodalherren
#1
Mar27-13, 05:57 PM
P: 331
1. The problem statement, all variables and given/known data

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.

2. Relevant equations



3. 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? :)
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Mark44
#2
Mar27-13, 06:52 PM
Mentor
P: 21,284
Quote Quote by Feodalherren View Post
1. The problem statement, all variables and given/known data

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.

2. Relevant equations



3. 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.
Quote Quote by Feodalherren View Post

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
#3
Mar27-13, 08:36 PM
P: 331
Thank you I got it now! :)


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