Cost minimization while retaining volume
I need to design a dumpster for my final calculus project. The main idea is that I find a dumpster, measure it, and retain the basic shape and construction when redesigning it. I must also keep the volume constant.
The problem would be a heck of a lot easier if it was just a rectangular prism, but it's a trapezoidal prism. The cost for the sides, front, and back is $0.70 cents per square foot (including any cuts or folds). The base costs $0.90 per square foot. The lid costs $50.00 regardless of dimensions. Welding, for joints, costs $0.18 per foot. This would be for side lengths, etc.
I know that I will have to use partial differentiation, but I'm at a loss as to how to start this and the general idea with the process of this problem. I can't use a rectangular prism instead of the trapezoid. Any help or explanations are greatly appreciated.