A packaging company is going to make closed boxes, with square bases, that hold 512 cubic centimeters. What are the dimensions of the box that can be built with the least material?

M=2b^2+4bh, V=hb^2, V=512 so

hb^2=512

h=512/b^2 using this value of h in the material function ...

M=2b^2+4b (512/b^2)

M=2b^2 + (2048b/b^2)

M=2b^2+2048/b

M = (2b^3+2048) / b

dM/db = (6b^3-2b^3-2048) / b^2

dM/db = (4b^3-2048) / b^2

dM/db=0 when 4b^3=2048

b^3=512

b=8cm, and since h=512/b^2

h=512/64=8cm

Therefore the box that uses the least material is a perfect cube with sides of 8cm.