>In regards to Greg's answer, I don't think it's that simple because the >problem >isn't simply to minimize area, it's to minimize cost. In fact, since the cost >of the lateral area is less, it's reasonable to assume a contrary >strategy...lower the area of the base and use a larger perimeter. > >Here's where I got >A=area of base P=perimeter of base h=height of solid > >The cost function c(A,P,h) = .01Ph+.05A > >Since h = 355/A > >C(A,P) = 3.55P/A + .05A > >And I see no relationship between A and P to put this in terms of a single >variable.
For any given fixed value of A, C(A,P) is minimized when P is as small as possible. That's why the regular hexagon is best for any value of A you pick.