APPLIED PROJECT Discs cut from squares Discs cut from hexagons THE SHAPE OF A CAN In this project we investigate the most economical shape for a can. We rst interpret this to mean that the volume V of a cylindrical can is given and we need to nd the height h and radius r that minimize the cost of the metal to construct the can (see the gure). If we dis- regard any waste metal in the manufacturing process, then the problem is to minimize the sur- face area of the cylinder. We solved this problem in Example 3.7.2 and we found that h = 2r; that is, the height should be the same as the diameter. But if you go to your cupboard or your supermarket with a ruler, you will discover that the height is usually greater than the diameter and the ratio 11/ r varies from 2 up to about 3.8. Let's see if we can explain this phenomenon. 1. The material for the cans is cut from sheets of metal. The cylindrical sides are formed by bending rectangles; these rectangles are cut from the sheet with little or no waste. But if the top and bottom discs are cut from squares of side 2r (as in the gure), this leaves consider- able waste metal, which may be recycled but has little or no value to the can makers. If this is the case, show that the amount of metal used is minimized when h i=~255 r 11' 2. A more efcient packing of the discs is obtained by dividing the metal sheet into hexagons and cutting the circular lids and bases from the hexagons (see the gure). Show that if this strategy is adopted, then h 4.5 =z2_21 r T 3. The values of h/r that we found in Problems 1 and 2 are a little closer to the ones that actually occur on supermarket shelves, but they still don't account for everything. If we look more closely at some real cans, we see that the lid and the base are formed from discs with radius larger than r that are bent over the ends of the can. If we allow for this we would increase h/r. More sipnicantlv. in addition to the cost of the metal we need to