Question

A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in.

Please help!

Answer 1

Explanation:

x = the length the base side

y = the height

12 sides = 2(4x) + 4(y) = 8x + 4y = 144

Divide by 4

Total sides = 2x + y = 36

y = 36 - 2y

The height y = 36 - 2x

Volume = yx²

Replace y with 36 - 2x

V = (36 - 2x)(x²) = 36 x² - 2x³

V = 36 x² -2x³

Maximum Volume

d/dx(36 x² - 2x³) = 72x - 6x²

= 6x (12 - x) = 0

x = 0 or x = 12

The side of the base is

x = 12

The max V = 12.12.12 = 12 = 1728 in³