Tin Can Generalization Project: The tin can of minimum cost in is not necessarily the

Chapter 8, Problem 16

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Tin Can Generalization Project: The tin can of minimum cost in is not necessarily the one with minimum surface area. In this problem you will investigate the effects of wasted metal in the manufacturing process and of overlapping metal in the seams. a. Assume that the metal for the ends of the can in costs k times as much per square centimeter as the metal for the cylindrical walls. Find the value of k that makes the minimum-cost can have the proportions of the normal can. Is it reasonable in the real world for the ends to cost this much more (or less) per square centimeter than the walls? Explain. b. Assume that the ends of the normal tin can in are cut from squares and that the remaining metal from the squares is wasted. What value of k in part a minimizes the cost of the normal can under this assumption? Is the can that uses the minimum amount of metal under this assumption closer to the proportions of the normal can or farther away? c. The specifications require that the ends of the can be made from metal disks that overhang by 0.6 cm all the way around. This provides enough overlap to fabricate the constructed cans top and bottom joints. There must also be an extra 0.5 cm of metal in the cans circumference for the overlap in the vertical seam. How do these specifications affect the dimensions of the minimum-area can in 15?