SOLUTION: 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 a

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Question 1127032: 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.)
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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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.)
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Simply saying,  the post is defective.


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Had this post be presented ideally  (as it should be),  it would be a  BRILLIANT  problem.




Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The formula for the volume of a cone is

V+=+%281%2F3%29%28pi%29%28r%5E2%29%28h%29

The diameter of the base is 4, so the radius r is 2; the height h is 6. So h is 3 times r.

As the tank fills, the height (depth of the water) is always 3 times the radius (of the surface of the water). So h = 3r, or r = h/3.

We are given the rate at which the water level is rising (dh/dt); so we want our volume formula to be in terms of h alone. So

V+=+%281%2F3%29%28pi%29%28%28h%2F3%29%5E2%29%28h%29+=+%28%28pi%29h%5E3%29%2F27

Find the derivative:

dV%2Fdt+=+%28%28%28pi%29h%5E2%29%2F9%29%28dh%2Fdt%29

The given height is 2m, which is 200cm; dh/dt is given as 20cm/min. Evaluate the derivative with those values.

dV%2Fdt+=+%28%28%28pi%29%28200%5E2%29%29%2F9%29%2820%29+=+279252 to the nearest whole number.

The water volume is increasing at a rate of 279,252 cm^3/min while 10,500 cm^3/min is leaking out; that means the rate at which the water is being pumped into the tank is 279,252+10,500 cm^3/min, or 289,752 cm^3/min.