Imagine that you are an engineer for a soda company, and you are tasked with finding the most economical shape for its aluminum cans. You are given a set of constraints: the can must hold a volume V of liquid and be a cylindrical shape of height h and radius r, and you need to minimize the cost of the metal required to make the can. a) First, ignore any waste material that is discarded during the manufacturing process and just minimize the total surface area for a given volume V. Using this constraint, show that the optimal dimensions are achieved when h = 2r. The formula for the volume of a cylinder is V = mh. The formula for the lateral area of a cylinder is L = 2rh. b) Next, take the manufacturing process into account. Materials for the cans are cut from flat sheets of metal. The cylindrical sides are made from curved rectangles, and rectangles can be cut from sheets of metal with virtually no waste. However, when the disks of the top and bottom of the can are cut from flat sheets of metal, there is significant waste material. Assume that the disks are cut from squares with side lengths of 2r, so that one disk is cut out of each square in a grid. Show that in this case the amount of material is minimized when: h 8 -=-=2.55 c) It is far more efficient to cut the disks from a tiling of hexagons than from a tiling of squares, as the former leaves far less waste material. Show that if the disks for the lids and bases of the cans are cut from a tiling of hexagons, the optimal ratio is: h 43 === 2.21 r Hint: The formula for the area of a hexagon circumscribing a circle of radius r is A = 61-2 d) Look at a variety of aluminum cans of different sizes from the supermarket. Which models from problems a-c best approximate the shapes of the cans? Are the cans actually perfect cylinders? Are there other assumptions about the manufacture of the cans that we should take into account? Do a little bit of research, and write a one-page response to answer some of these questions by comparing our models to the actual dimensions used.