Calculus help, word problems are my weakness

Question

A rectangular solid with a square base has a surface area of 100 square cm. Find the dimensions that will result in a solid with maximum volume.

Answer 1

The Volume = (x)(x)(h) = x²h

But the surface area = xh + xh + xh + xh + x² + x² so 100 = 4xh + 2x² or 50 = 2xh + x² so h = (50 - x²)/2x

So since V = x²h we can say V(x) = x²(50 - x²)/2x (replacing h with "(50 - x²)/2x") or:

V(x) = x(50 - x²)/2

V(x) = 25x - 1/2x³

Th find extrema (max/min), make V'(x) = 0

V'(x) = 25 - 3/2x²

0 = 25 - 3/2x²

25 = 3/2x²

x² = 25(2)/3

x = √(25(2)/3) = 5√(2/3) = 5/3√6

Plugging this value in for x into h = (50 - x²)/2x we also get h = 5/3√6

Answer 2

Something must be missing from this problem. You'd get maximum volume with maximum height, or an infinite height. Dimensions would be infinity x 10cm x 10cm

So what might be missing from the problem is that the solid has some limits on material to create it. A cube is generally the solution for some types of these problems, which would mean 10cm x 10 cm x 10 cm, but that assumes 10 cubed or 1000 cubic centimeters as the material to make the solid.

Or make the volume of the rectangular solid some arbitrary number, call it V, as if there is some limit on the material to make the solid. V=100h 100=the side of the base squared or 10 cm x 10 cm.

V(h)=100h h=V/100 then the dimensions of the solid are 10x10xV/100.

But this is supposedly a calculus problem, so maybe there's something else not given that would have made it into a more typical calculus problem.