# What is the radius of the one with maximum volume?

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?

• Volume of a R-C-Cylinder 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