Cutting Equal StripsDate: 09/08/2005 at 09:21:33 From: george Subject: Dimensions Given five 4 ft by 8 ft rectangular sheets of plywood, you are assigned to cut strips that are 7 1/2 inches wide and 8 feet long. Each saw cut eliminates 1/16 inch of the plywood as sawdust. You are not able to glue any strips together. Ignoring the sawdust that results, show how many 7 1/2 inch by 8 foot strips you will get from the five sheets of plywood and give dimensions of the leftover strips of plywood. I have tried to work it out many different ways but it all comes out wrong. Date: 09/08/2005 at 10:02:33 From: Doctor Peterson Subject: Re: Dimensions Hi, George. Let's simplify the numbers so you can get a sense of what's happening, and then you can apply that understanding to solving the actual problem. Suppose you want to cut 3-inch strips from a 14-inch wide board, but you have a really bad saw that turns 1 inch of wood into sawdust. You cut 3 inches from the left side and take out the next inch, like this: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+ <---------> <-> strip 1 saw So the next strip will begin at what was the 4-inch mark on the original board, and take up the next 3 inches: +---+---+---+---+---+---+---+---+---+---+---+---+---+---+ <---------> <-> <---------> <-> strip 1 saw strip 2 saw Repeat again, and we have +---+---+---+---+---+---+---+---+---+---+---+---+---+---+ <---------> <-> <---------> <-> <---------> <-> strip 1 saw strip 2 saw strip 3 saw So we have a 2-inch strip left that is too small to make another 3- inch strip from. How could we have calculated this? Well, each strip uses up not just 3 inches, but 4 inches of wood; so we can divide 14 by 4 to find how many times we can repeat the process--that is, in cutting three times, we used up 3 times 4 inches of wood, counting both the strips we made and the sawdust that was produced. There is just one little detail to worry about. What if we had had a 15-inch board to start with, so that the last strip was exactly 3 inches wide, and we didn't have to make the last cut? +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ <---------> <-> <---------> <-> <---------> <-> <---------> strip 1 saw strip 2 saw strip 3 saw strip 4 Dividing 15 by 4 gives 3 with a remainder of 3, and we'd have to recognize that the remainder is enough to leave one more strip. There's a neat trick I can see that would allow us always to find the number of strips we can make by dividing; it involves pretending that the board started out 1 inch wider than it is, so that we would need that one last cut and have nothing left. But you don't need that trick in order to solve the problem. If you need more help, please write back and show me how far you got in solving the real problem. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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