An open-top shipping crate with square bottom and rectangular sides is to hold 32 in^3 and requires a minimum amount of cardboard. Find the most economical directions.

Answer 1

volume: s^2 *h where s is the bottom side length, h is height.
area: s^2+ 4sh

but 32=s^2 h so h=32/s^2

darea/ds=d(s^2 + 4*32/s)
2s-128/s^2=0
solve for s, then h

check to see that area is minimum.