If 2000 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

I assume you are cutting square tabs from the corners of a square sheet of material, and folding up the sides. If so, then if the tabs have side h, then if the base has side length x, and the height is h, then
(x-2h)^2 + 4xh + 4h^2 = 2000
x^2 + 8h^2 = 2000
so, x^2 = 2000-8h^2
The volume of the box is thus
v = x^2h = h(2000-8h^2)
so to find max volume, find where dv/dh = 0

Answer 2

So, no lid, right?

If so, let each side of the square base be x cm
let the height be y cm
surface area = x^2 + 4xy = 2000
4xy = 2000 - x^2
y = (2000 - x^2)/(4x) = 500/x - x/4

Volume = x^2 y
= x^2(500/x - x/4) = 500x - x^3 /4
d(Volume)/dx = 500 - (3/4)x^2 = 0 for a max volume
(3/4)x^2 = 500
x^2 = 2000/3 or 6000/9
x = √6000 / 3 = 20√15/3

since V = 500x - x^3 /4, plug in your value of x
I will let you do all that arithmetic, the "math" part of the question is done

Answer 3

I took the question literally as "2000 square centimeters of material is available", so no waste.