Your iron works has contracted to design and build a 500-cubic foot, square based, open topped, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible. What dimensions do you tell the shop to use?
Suppose the base is x feet and the height is y feet.
It is an open box, so its surface has 5 sides. To total area of the surface is ...
This shows how to find dimensions to minimize weight.