Water is leaking out of an inverted conical tank at a rate of 10,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. the tank has height 6 m and the diameter at the top is 4 m. if the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (round your answer to the nearest integer.)

The tank has a volume of , where is its height and is its radius.

At any point, the water filling the tank and the tank itself form a pair of similar triangles (see the attached picture) from which we obtain the following relationship:

The volume of water in the tank at any given time is

and can be expressed as a function of the water level alone:

Implicity differentiating both sides with respect to time gives

We're told the water level rises at a rate of at the time when the water level is , so the net change in the volume of water can be computed:

The net rate of change in volume is the difference between the rate at which water is pumped into the tank and the rate at which it is leaking out:

We're told the water is leaking out at a rate of , so we find the rate at which it's being pumped in to be

powerful computers now, not then

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