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Donkey Grazing Half a Field

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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)?
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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

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/
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Associated Topics:
High School Calculus
High School Conic Sections/Circles
High School Geometry

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