Mark H. answered • 12d

Tutoring in Math and Science at all levels

V = π * **r**^{2} * h, and r + h = 12

You have 2 independent variables, so use both equations to develop a formula for V. It seems best to use substitution for h, to keep the equation relatively simple....So:

r + h = 12

rearrange to get: h = 12 - r

Substitute in the 1st equation to get:

V = π * r^{2} * (12 - r)

Expand:

V = 12πr^{2 }- πr^{3}

To find a maximum (or minimum), the typical method is to find the 1st derivative and set it equal to zero.

V' = 24πr - 3πr^{2}

Set equal to 0 and solve for r:

24πr - 3πr^{2 }= 0

8r - r^{2 }= 0

r = 0, 8 (Interpretation: 0 gives you the minimum volume, and 8 gives you the maximum)

CHECK this by plugging into the original formula and then trying values on either side.