We want to build a box whose base is square, has no top and will enclose 100 m^3. determine the dimensions of the box that will minimize the amount of material needed to construct the box.

Let the height of the box be h and the length of a side of the base be s.
Then there are four sides of area hs and one bottom of area s^2.
The volume of this box is then V=(s^2)(h) = 100 m^3.  We want to minimize the surface area.  The surface area is A = 4sh+s^2.
We can solve the first equation for h and substitute the result into the second equation, to derive a function s alone.  If V=hs^2, then h=V/s^2.
Substituting this into A = sh+s^2, A=s(V/s^2)+s^2.

We must minimize this function.  First, simplify it to V/s + s^2.  Then differentiate it with respect to s:

dA/ds = -V/s^2+2s.

Set this = to 0 and solve for s:  2s=V/s^2.  Mult. both sides by s^2:

2s^3=V.  Then s^3 = V/2, and s = cube root of V/2.

The dimensions required for minimum surface area are (cube root of V/2)^2 (area of the base) times h, where h is the height and is equal to V/s^2, or V/[cube root of V/2]^2.