A rectangular solid (with a square base) has a surface area of 121.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.

Question

A rectangular solid (with a square base) has a surface area of 121.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.

= cm (smallest value)
= cm
= cm (largest value)

Answer 1

Let x = the side of the square base and y = height

Let V = volume = x²y

The surface area = 2x² + 4xy = 121.5

 

Solve the surface area equation for y = (121.5 - 2x²)/4x and substitute this into the volume equation

 

V = x²(121.5 - 2x²)/4x = x(121.5 - 2x²)/4

 

the derivative of V with respect to x = (x/4)(-2x) + (1/4)(121.5 - 2x²) = 0 at the maximum

 

(1/4)(-4x² + 121.5 - 2x²) = 0'

6x² = 121.5

x² = 20.25

x = 4.5

y = (121.5 - 40.5)/(4 * 4.5) =81/18 = 4.5

 

The the rectangular solid with maximum volume for given surface area is, in fact, a cube with side 4.5. 

 

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