SOLUTION: Water is being pumped into a conical tank at the rate of 100 ft3/min. The height of the tank is 20 ft and its radius is 5ft. How fast is the water level rising when the water heig

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Question 650834: Water is being pumped into a conical tank at the rate of 100 ft3/min. The height of the tank is 20 ft and its radius is 5ft. How fast is the water level rising when the water height is 10ft?
Answer by htmentor(1343) About Me  (Show Source):
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Water is being pumped into a conical tank at the rate of 100 ft3/min. The height of the tank is 20 ft and its radius is 5ft. How fast is the water level rising when the water height is 10ft?
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The volume of a cone, V+=+%28pi%2F3%29r%5E2%2Ah
We can express r in terms of h: r = h/4
So we have an equation for the volume in terms of the height only:
V+=+%28pi%2F3%29%28h%2F4%29%5E2%2Ah+=+%28pi%2F48%29h%5E3
The rate of change of volume, dV%2Fdt+=+%28dV%2Fdh%29%28dh%2Fdt%29+=+100+ft%5E3%2Fmin
dV%2Fdh+=+%28pi%2F16%29h%5E2
The rate of change of the water height, dh%2Fdt, at h=10 is:
dh%2Fdt+=+%28100%29%2F%28%28pi%2F16%29%2A10%5E2%29+=+16%2Fpi
So the rate of rise at h=10 ft = 5.093 ft/min