# Water is leaking out of an inverted conical tank at the rate of 10,000cm^3/min cm/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 20 cm/min when the height of the water is 2m, how can I find the rate at which water is being pumped into the tank.?

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#### Explanation

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(see below for solution method)

#### Explanation:

Start by ignoring the leakage and determine the rate of inflow required to achieve the specified rate of height (depth) of water increase.

Later we'll use the fact that

**Actual inflow rate
= Inflow Rate for Increased Depth + Leakage Rate**

For the given cone the ratio of **r** adius to **h** eight is

so

The formula for the volume of a cone:

We are interested in the change in Volume with respect to time and note that

Using the value we've already calculated for

we get:

or roughly

This is the Inflow Rate Required to Cause Height Increase and

ignores the Rate of Leakage

The Actual Inflow Rate needs to be the sum of these two:

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