The design of a digital box camera maximizes the volume while keeping the sum of the dimensions at 6 inches. If the length must be 1.5 times the height, what should each dimension be?

Hint: Let x represent one of the dimensions, then define the other dimensions in terms of x.

H = x, L = 1.5 x;

L + W + H = 6

x + 1.5 x + W = 6

W = 6 - 2.5 x

Dimensions of the camera in terms of x:

x, 1.5 x, 6 - 2.5 x.

V = L x W x H

V = x * 1.5 x * (6 - 2.5 x) = 9 x² - 3.75 x³

V ' = 18 x - 11.25 x² (V max is when V' = 0)

18 x - 11.25 x² = 0

x (18 - 11.25 x) = 0

11.25 x = 18

x = 18 : 11.25

x = 1.6

The dimensions are:

L = 2.4

W = 2.0

H = 1.6