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Circular Field, Cow, and Length of Rope

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 
answers.  Point your web browser at the address at the bottom of this 
message, and go to the High School Geometry section. 

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

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.

Solve that, and you'll have your answer.

-Doctor Ken,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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