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

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