The net for this box would be:

The volume of the box is V = x²y

We are told that the surface area of the box is 64 cm². The area of the base of the box is x² and the area of each side is xy, so the area of the base plus the area of the 4 sides is given by:

x² + 4xy = 64 cm²

Solving for y gives:

So the volume can be rewritten:

Now

and this is zero when x = ± 8/√3 ≈ ± 4.62.

(Note: The negative case has no practical meaning.)

Is it a maximum?

and this is negative when x is positive. So it is a MAX.

So the dimensions of the box are:

Base 4.62 cm × 4.62 cm and sides 2.31 cm.

The maximum possible volume is

V = 4.62 × 4.62 × 2.31 ≈ 49.3 cm³

Check: Area of material:

x² + 4xy = 21.3 + 4 × 4.62 × 2.31 = 64

Checks OK.