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### Circular Field, Cow, and Length of Rope

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Date: 9/11/96 at 13:57:25
From: Anonymous
Subject: Circular Field, Cow, and Length of Rope

If a cow is anchored by a chain to the perimeter of a circular field
of grass with the other end of the chain tied around its neck; given
that the field has a diameter of 10 feet, how long does the chain have
to be for the cow to eat half the grass? Is there a ratio between the
radius and the length of chain? Please detail the explanation as much
as possible. Thank you.
```

```
Date: 9/11/96 at 16:36:42
From: Doctor Ken
Subject: Re: Circular Field, Cow, and Length of Cow's Rope

Hello -

It just so happens that we've answered cow questions like this before
(complete with pictures!), so you can browse our archives to see the
message, and go to the High School Geometry section.

However, these problems are a little different from the one you
propose.

Let L be the length of the chain.  Then the area that the cow can
graze is given by

100 Pi - (10-L)^2 Pi.   (the whole area minus the inner circle)

If this is half of the area of the entire field, then it is equal to
50 Pi, so we get

100 Pi - (10-L)^2 Pi = 50 Pi.

-Doctor Ken,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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