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

think about the oscillation of the toy shown in the video. when the toy is at the highest point of its motion, its velocity is zero and its acceleration is negative.