Cow Grazing in Circles

Date: Thu, 3 Nov 1994 21:27:35 -0500 (EST)
From: Matthew T Phillips
Subject: One Heck of a Problem

I've got a good one for you.

        A cow is tied to a 100 ft. rope (stay with me - this _is_ 
interesting).  His rope is tied to a pole in the center of a circle of 
radius 50 ft.  This circle has a ten foot opening, out of which the cow 
can walk and graze.  The cow has the resulting grazing area: (excuse the 
crudity of my drawing.  What's the area of grazing?
     _______
   _/       \_      The difficulty of the problem is the small wedge
  /           \     on each corner of the circle.
 |    /   \    |
  \--|     |--/     This problem came from the American Society of Civil 
      \___/         Engineers.

I know this is a difficult problem, but if you find a solution, please 
respond to me.  Thanks.

          phillips@oven.ccds.charlotte.nc.us    

From: Dr. Ken
Date: Fri, 4 Nov 1994 11:18:06 -0500 (EST)

Hello Matthew!

Here's what my first instinct would be on this one.  I assume that if you're
looking at this problem and find it interesting, you've also seen simpler
problems of the same type.  For instance, what if the structure he was tied
to were a square, and not a circle?  Then we would be able to break the
grazing area down into chunks, and add up the pieces.  If you haven't worked
any problems like this before, try some.

Then what happens if you tie the cow to a pentagon of about the same size?
A hexagon?  I think it will be helpful to actually write out (without
condensing first, then in closed form) the sums you would get using these
more regular shapes, and then see if you can find a pattern, i.e. see if it
looks like the sum is approaching a certain number (it is) as the number of
sides on the structure gets big (i.e., its shape approaches the shape of a
circle).

Let us know how it comes out!

-Ken "Dr." Math "Williams"