Given a right circular cylinder whose radius increases at a constant rate and whose altitude is a linear function of the radius. Also, given that the altitude is increasing at a rate three times that of the radius. The volume is increasing at a rate of 1 cubic feet per second when the radius is 6 feet. When the radius is 36 feet the volume is increasing at a rate of cubic feet per second. Find the value of the integer .
The following diagram illustrates the setup:
Since the altitude (which we denote ) is a linear function of the radius and increases three times as quickly, we have
When , we are given . Thus, we solve for and get .
When , we have
So, since implies and
Solving for we obtain,
Then, when we have