Goat on a Rope

Date: 8/6/96 at 9:43:2
From: Dave Nicholls, Information Analyst
Subject: Goat Grazing from Edge of Circle

Hi...

I've got a simple question that I've never been able to work out. 
I'll set the scene:

A circular field, diameter X, has a white picket fence around its 
edge.  A goat's tether rope of length Y is attached to one of the 
these pickets. How long does the rope need to be so that the goat can 
graze exactly half the field?

-Dave

Date: 8/6/96 at 16:42:1
From: Doctor Anthony
Subject: Re: Goat Grazing from Edge of Circle

I shall use my own terminology for this problem, but here is the 
calculation:

Since one cannot draw a decent diagram with ascii, I suggest you
draw the following diagram on a piece of paper and refer to it while
I go through the working.  Draw a circle with suitable radius r.
Now take a point C on the circumference and with a slightly larger
radius R draw an arc of a circle to cut the first circle in points 
A and B. Join AC and BC. Let O be the centre of the first circle of
radius r. Let angle OCA = x (radians). This will also be equal to
angle OCB. The area we require is made up of a sector of a circle
radius R with angle 2x at the centre, C, of this circle, plus two
small segments of the first circle of radius r cut off by the chords
AC and BC.

Area of sector of circle R is (1/2)R^2*2x = R^2*x

area of two segments = 2[(1/2)r^2(pi-2x) - (1/2)r^2sin(pi-2x)]
                     = r^2[pi - 2x - sin(2x)]

We also have R = 2rcos(x)   so R^2*x = 4r^2*x*cos^2(x)

We add the two elements of area and equate to (1/2)pi*r^2

 4r^2*x*cos^2(x) + r^2[pi-2x-sin(2x)] = (1/2)pi*r^2   divide out r^2

     4x*cos^2(x) + pi - 2x - sin(2x)  = (1/2)pi
 
 4x*cos^2(x) + (1/2)pi - 2x - sin(2x) = 0

We must solve this for x and we can then find R/r from R/r = 2cos(x)

Newton-Raphson is a suitable method for solving this equation, using
a starting value for x at about 0.7 radians

The solution I get is x = 0.95284786466 and from this
                  cos(x) = 0.579364236509

and so finally  R/r = 2cos(x) = 1.15872847


-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 05/25/97 
From: Doctor Sarah

You'll find more answers to interesting problems about tethering and 
grazing, some with illustrative diagrams, by searching the Dr. Math 
archives for the words "goat" and "cow" (just the word, not the 
quotes).  

Here are two more answers by Dr. Anthony:

"Coordinate Geometry - Goat in a Circular Field":

 http://mathforum.org/dr.math/problems/booth8.22.96.html   

"The Goat in the Field Problem"

 http://mathforum.org/dr.math/problems/miller5.24.97.html   

-Doctor Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/