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# Water is leaking out of an inverted conical tank at a rate of 6700.0 cm³/min at the same time that water is being pumped...

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Water is leaking out of an inverted conical tank at a rate of 6700.0 cm³/min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 15.0 m and the diameter at the top is 4.5 m. If the water level is rising at a rate of 23.0 cm/min when the height of the water is 2.0 m, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

May 3, 2022

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Sketch of solution:

Suppose the rate at which water is being pumped into the tank is k cm^3 / min. Then the net rate of increase of volume is (k - 6700) cm^3 / min.

You can do what I did in the previous problem (considering the largest cross-section) to find the radius of the water surface when the height is h cm. Draw a diagram to help you visualize the problem. When you have that, you can express the volume V entirely in terms of h. Differentiate both sides with respect to time and then you can relate the rate of change in volume to the rate of change in water level. That rate of change in volume when h = 2 is exactly (k - 6700) cm^3 / min.

Please tell me if you encounter problems solving it according to my sketch.

May 3, 2022