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.
