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Donkey Grazing Half a Field


Date: 08/08/97 at 09:27:13
From: Tim Steele
Subject: Donkey in Field

Can you solve this for me?

A (zero-dimensional) donkey is attached by a (one-dimensional) rope
to a point on the perimeter of a (two-dimensional) circular field.

How long should the rope be (in terms of the radius of the field)
so that the donkey can reach exactly half the field (eat half
the grass)?


Date: 08/08/97 at 12:38:24
From: Doctor Anthony
Subject: Re: Donkey in Field

This problem comes up from time to time in the guise of the length of 
rope required to tether a goat (instead of a donkey) on the boundary 
of a circular field such that the goat can eat exactly half the grass 
in the field.

Here is a diagram you should refer to 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
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Associated Topics:
High School Calculus
High School Conic Sections/Circles
High School Geometry

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