optimization from hell

[englishbum]

I have a really tough question and need like emergency help and aid...

a juice manufacturer is studying the most economical shape to use for a beverage container. Each unit will contain 335cm^3 of juice. The manufacturer is considering a cylinder versus a rectangular prism with a comfortable hand-held depth of 4cm. Which method od packaging, the can or the juice box, will use the minimum amount of materials?

Thanks in advance for the help....tonnes o'thanks

[StatusX]

If I understand the problem, you have a rectangluar prism and a cylinder, both of which have a certain constrained hieght, you are given the necessary volume, and you need to see which has less surface area. This isn't hard at all, and doesn't even involve calculus. Just keep in mind that the two unknown sides of the prism must be equal by symmetry, then find these sides and the radius, and see which shape has a smaller surface area.