Question

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?

Answer 1

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.