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# Find the relative rate of change of volume to rate of change of height in a water tank

Consider a water tank shaped like a hemisphere with a radius of 10 feet. At time let,

Find the rate of change of the volume relative to the rate of change of the height () when .
If water is flowing into the tank at a constant rate of cubic feet per second, find when .

For this problem we will consider the graph of the following function (the hemisphere tank and water will then be obtained as solids of revolution of this graph about the -axis):

First, we find a formula for the volume of the water in the tank as a function of . We do this by considering the solid of revolution (for a review of solids of revolution see these exercises) of about the -axis:

Differentiating with respect to we then have,

So, when feet we have

Next, we are given cubic feet per second. We know from above, . So,

Then, to get in terms of we evaluate,

Thus,

So, if , we have