Toward a Quantitative Human Ecology of Soup Cans

  1. Cylindrical soup cans are to be made to hold a fixed volume V. To save resources, you need would like to design a can that will hold V, but will use the least metal. What is the ratio of the radius to the height for the can that uses the least metal? (We've already done this problem in class, but used 1000 for V.)

  2. Cylindrical soup cans are to be manufactured to hold a fixed volume V. There is no waste in cutting the metal for the sides of the can, but the circular endpieces will be cut from a square, with the corners wasted. Find the ratio of the radius to the height for the most economical can. Is your answer bigger or smaller than your answer for part 1? Explain.

  3. Suppose that you don't need to worry about the wasted corners any more; you've convinced the plant operators that you should save the corners and re-melt them and use them again. So they're not really wasted. However, your engineers tell you that for better stability, you'll need make the top and the bottom of the cylinder three times as thick as the sides. Find the ratio of the radius to the height for the can that uses the least material. Is your answer bigger or smaller than your answer for part 1? Explain.

  4. Suppose you want to break with tradition and make your soup cans conical instead of cylindrical. Find the ratio of the radius to the height for the cone that holds a fixed volume V but uses the least material.

  5. Now suppose that the soup can material costs C1 per square centimeter. Also, assume a cost C2 per centimeter of "seam" along the top and bottom of the cylindrical can. Find the ratio of the radius to the height for the least expensive can. You will probably want to use wolfram alpha for this. Discuss your result. Does the ratio change as you would expect if C1 is much larger than C2? What if C2 is much larger than C1?

  6. (Optional:) You now need to build a structure to hold your soup cans before the truck comes to take them to natural food stores and co-ops. Not having taken any architecture classes, you decide on a very simple design: a large piece of metal folded down the middle to form a long V. This V will then form a long triangular shelter for your soup cans. (To visualize this shape, take a piece of paper and fold it in half. Unfold it and stand it up on its ends.) How should you fold the metal so that your shelter will be able to hold the most soup cans? I.e., you want to maximize the volume of the building. Assume that the length of your metal before folding is L and the width is W. (Hint: There are many ways to solve this problem. I think the easiest is to express the volume of the structure in terms of the angle of folding. Then take the derivative of V with respect to this angle and minimize.)