What is the maximum volume of a closed box with a square base which can be made by bending the material? The box is to have a surface area of 100in^2.

Answer 1

If the final box has bases x^2 and height h, then

v = x^2 h

and we also have

x^2 + 4xh = 100, so
h = (100-x^2)/4x

That means

v = x^2 (100-x^2)/4x
= x/4 (100-x^2)
= 25x - 1/4 x^3

dv/dx = 25 - 3/4 x^2

maximum volume is achieved when dv/dx=0.