Is to be constructed for use on padre island beaches. it is to have a back, two square sides, and a top. if 96 square feet of canvas are available, find the length of the shelter for which the volume inside is maximized assuming all the canvas is used?

Area of the canvas: 96 = 2 · W² + 2 · W L,

where W is the width and L is the length of a back and a top side. Two square sides have area W².

2 W L = 96 - 2 W² / : 2

W L = 48 - W²

L = (48 - W²) / W.

The inside volume of the shelter:

V = L · W · W = L · W²

V = W² · (48 - W²) / W;

V = 48 W - W³

We have to find the derivative:

V ' = 48 - 3 W²

48 - 3 W² = 0 (for V max.)

3 W² = 48

W² = 48 : 3

W² = 16; W = √16; W = 4 ft.

L = (48 - 16) / 4 = 32 / 4 = 8 ft.

Answer: The length of the shelter for which the volume inside is maximized is 8 ft.