Step-by-step explanation:
Let h be the height of box and a be the side of base square.
We have 108 ft² material.
Surface area of trash bin = 108 ft²
Surface area of trash bin = a² + 4 ah
a² + 4 ah = 108
4ah = 108 - a²
[tex]h=\frac{108-a^2}{4a}[/tex]
Volume of trash bin, V = a²h
[tex]V=a^2\times \frac{108-a^2}{4a}=27a-0.25a^3[/tex]
At maximum volume differentiation of volume will be zero
[tex]dV=27da-0.25\times 3a^2da\\\\0=27da-0.75a^2da\\\\a^2=36\\\\a=6ft[/tex]
Surface area of trash bin = a² + 4 ah = 108
36 + 4 x 6 x h = 108
h = 3 ft
Dimension of trash bin is 6 ft x 6 ft x 3ft .
