Question

A canvas wind shelter like the one at right is to be built for use along parts of the Guadalupe River. It is to have a back, two square sides, and a top. If 147/2 square feet of canvas is to be used in the
construction, find the depth of the shelter for which the space inside is maximized assuming all the canvas is used.

Answer 1

A = 2 a² + 2 a b 
2 a² + 2 a b = 147/2
a² + a b = 36.75
a b = 36.75 - a²
b = ( 36.75 - a² ) / a
V = a² * b
V = a² * ( 36.75 - a² ) / a
V = 36.75 a - a³  ( maximum of the volume is when V` = 0 )
V` = 36.75 - 3 a²
36.75 - 3 a² = 0
3 a² = 36.75
a² = 36.75 : 3
a² = 12.25
a =√12.25
a = 3.5 ft
b = 24.5 / 3.5 = 7 ft.
Answer: The depth of the shelter for which the space inside is maximized is 3.5 ft.