# Water leaking out of a cone

by Feodalherren
Tags: cone, leaking, water
 P: 216 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? :)
Mentor
P: 20,433
 Quote by Feodalherren 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 by Feodalherren 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? :)
 P: 216 Thank you I got it now! :)

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