Question

If 2,700 cm2 of material is available to make box with square base and an open top, find the dimensions that give the largest possible volume of the box

Answer 1

If the box has base side x and height h, then assuming the material is also a square, we have
(x+2h)^2 = 2700
v = x^2 * h = -1/2 x^3 + 15√3 x^2
dv/dx = -3/2 x^2 + 30√3 x
dv/dx = 0 at x = 20√3, so v has a maximum of 6000√3
The box is 20√3 by 20√3 by 5√3