Another Grazing Cow
Date: Wed, 7 Jun 1995 12:53:17 -0500 Subject: A math riddle From: Ken Ryland A friend came by my office at work yesterday touting a math riddle that he claimed was insoluble (at least until the present). Perhaps you would like a crack at it. A man has a barn that is 20 ft by 10 ft. He tethers a cow to one corner of the outside of the barn using a 50-ft rope. What is the total area that the cow is capable of grazing? ------------------------- | | | | | | | | | | | |10 ft ------------------------- 20 ft \ \ 50 ft \ \ \ cow (you get the idea.) The smart alec who posed this question is Jack McVey. His e-mail address is email@example.com if you would like to send him a copy of your response. Hope to hear from you soon. Yours truly, Ken Ryland
Subject: Re: A math riddle From: Dr. Ken Date: Thu, 8 Jun 1995 15:02:24 -0400 (EDT) Hello there! Well, it seems to me that this problem is far from insoluble. In fact, I think that if I get you started a little bit and if you know how to do some integration, you'll be able to solve it yourself! Also, I made a nice sketch of the problem using The Geometer's Sketchpad: Basically, the shape of the grazing area is circular with a little chunk out of it. In my drawing, the barn looks like this: COW / / / / / / / / / / / / / / / / / / / ____________/ | | | | | | | | | | | | | | | | | | | | | | |____________| With this setup, the chunk that's missing from the grazing circle is in the lower left-hand corner. The right half of the grazing area, then, is just the area of the semicircle, which is 1250. To find the area of the remaining grazing area (including the barn, which you subtract later), you use integration. You see how the grazing area in the lower left-hand corner is the intersection of two circles? Well, you need to find that intersection point, and then integrate to find the area of the grazing area to the left of the intersection point, then the area to the right of the intersection point. I'm trying not to do too much for you here so that you can grab all the glory with your friend. If you want more help, please write back. Ken "Dr. Math"
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