Abox with a square base and no top is to have a volume of 32 cm3. find the least amount of material (surface area) needed to construct such a box.

The least amount of material will be required when the box is half a cube, so its surface area is half that of a cube with volume 64 in³. Such a cube has sides that are 4 in squares, so has an area of 4²×6 = 96 in². Half that is 48 in².

The least amount of material required for an open square box of volume 32 in³ is 48 in².

Let x represent the edge length of the square base. Then the height of the box is 32/x², and its surface area is

... A = x² + 4x(32/x²) = x² +128/x

The derivative of this is

... dA/dx = 2x -128/x²

and it will be 0 where the area is a minimum:

... 0 = 2x -128/x²

... 64 = x³ . . . . . multiply by x²/2 and add 64. Note this is the answer we had above.

... x = ∛64 = 4, and the area is

... A = 4² + 128/4 = 16 + 32 = 48 . . . . in²