This one has been kicking my rear for too long so any help is appreciated.
A box (top included) is to be made to hold exactly 216 cubic inches. It is to have a square base. Find the dimensions of the box that has minimum surface area.
The volume is 216 inches cubed...
You want to minimize the surface area. Key word here is minimize, so you must derive the surface area equation with respect to one of the sides.
Let's solve the surface area equation for one variable, say s.
Rearranging our prior equation for volume:
Plug that in for h in our SA equation.
Differentiate this with respect to s and set it equal to zero. Solve for s. Whichever value of s gives the minimum SA (plug s back into original SA equation) is your answer.