Consider all right circular cylinders for which the sum of the height and the circumference is 30 centimeters. What is the radius of the one with maximum volume?
2 Answers

Volume of a RCCylinder is defined as:
V = piR^2 H
The circumference is the basic eqeuation:
C = 2pi*R
According to the problem, C + H = 30
or
2pi*R + H = 30
H = 30  2pi*R
and volume becomes:
V = piR^2 H > V = piR^2 (30  2pi*R)
=30piR^2  2pi^2 R^3)
To find the one with maximum volume, take the derivative of the volume equation and set equal to zero(and solve for the required R):
V' = 60piR  6pi^2 R^2 = 0
6piR(10  pi R) = 0
6pi*R = 0
R = 0, which is not the answer we want...
OR
(10  pi * R) = 0
R = 10/pi cm

Right Circular Cylinders