# 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. (Optional:) It is more efficient to cut the circle for the lids from hexagons that are densely packed. See the image at this article on circle packing. If one does so, what is the ratio of the height to the radius for the optimal can?

4. 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.

5. 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.

6. (Optional:) 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?