A box with a square and a closed top must have a volume of 125 cubic inches. Find the dimensions of the box that minimize the amount of the material used.

If a box has a square base its volume will be:

V=hb^2 where h is the height ...

h=V/b^2 we are told that V=125 so

h=125/b^2 now for the surface area, which consists of the two bases for a total of 2b^2. It will also have four sides with a total area of 4 (bh) = 4bh so

A=2b^2+4bh, using h found above in this gives us:

A=2b^2+4b (V/b^2)

A=2b^2+4V/b

A = (2b^3+4V) / b, then taking the derivatives we can find the velocity of the area function.

dA/db = (6b^3-2b^3-4V) / b^2

dA/db = (4b^3-4V) / b^2

dA/db=0 when 4b^3-4V=0

b^3=V

b=V^ (1/3), since V=125

b=5in, and since h=V/b^2

h=125/25=5in

So the dimensions that will minimize the amount of material used to enclose a volume of 125in^2 is a 5in cube.

h=b=5in