If a total of 1900 square centimeters of material is to be used to make a box with a square base and an open top, find the largest possible volume of such a box.

Answer 1

maximize xyz subject to
xy + 2xz + 2yz = 1900

If it were a complete cube, max volume is where all faces are square.

In this case, missing the top, the area must be divided 1/3 to base and 2/3 to sides.

So, x=y=√(1900/3) and z=x/2
The box is 25.1661 x 25.1661 x 12.5831