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

we have,

Solving for we obtain,

Then, when we have