Question

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.

Answer 1

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