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Goat Problem

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The goat problem (or bull-tethering problem) considers a fenced circular field of radius a with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length L, and asks for the solution to various problems concerning how much of the field can be grazed.

GoatProblem

Tying a goat to a point on the interior of the fence with radius a=1 using a chain of length L=r, consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection

A=r^2cos^(-1)((d^2+r^2-R^2)/(2dr))+R^2cos^(-1)((d^2+R^2-r^2)/(2dR))-1/2sqrt((-d+r+R)(d+r-R)(d-r+R)(d+r+R))
(1)
GoatProblemPlot

Taking R=d=1 gives

A(r)=-1/2rsqrt(4-r^2)+r^2cos^(-1)(1/2r)+cos^(-1)(1-1/2r^2),
(2)

plotted above. Setting A=pi/2 (i.e., half of piR^2) leads to the equation

-1/2rsqrt(4-r^2)+r^2cos^(-1)(1/2r)+cos^(-1)(1-1/2r^2)=1/2pi,
(3)

which has numerical solution

r=1.15872847...
(4)

(OEIS A133731). Ullisch (2020, 2024) gives an exact expression for the solution using contour integrals.

Now instead consider tying the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius a. Assume that L<=api, so that the goat is not able to reach further around than the point on the fence opposite his starting point (Hoffman 1998, where we have replaced Hoffman's bull with a more prosaic goat). The goat may obviously graze inside the interior of a semicircle of radius L whose diameter is tangent to the fence. In addition, the goat may graze two areas on either side of the semicircle that have the fence as the inner boundary and a circle involute as the outer boundary. To find the area of this region, assume the fence is oriented so that the farthest point around the circumference that the goat can reach is at position (a,0). Now, note that the equation for a circle involute is given by

x(t)=a(cost+tsint)
(5)
y(t)=a(sint-tcost).
(6)

From geometry, the goat will transition between being radially bound and being bound by pulling tangent to the circle at the point where r^2=L^2+a^2, so

r^2=x^2+y^2
(7)
=a^2(1+t^2)
(8)
=L^2+a^2.
(9)

The last equality gives t=L/a for the transition parameter. The area of the involute portion that the goat can graze is then given by

A_(involute)=1/2int_0^(L/a)(xy^'-yx^')dt
(10)
=1/2a^2int_0^(L/a)t^2dt
(11)
=(L^3)/(6a).
(12)

Adding twice this area to the area of a semicircle of radius L then gives the total area which the goat can graze as

A=(piL^2)/2+(L^3)/(3a).
(13)
TetheredBullProblem

The grazable areas are illustrated above for a number of ratios of L/a. Note that the case L/a=pi forms a curve that resembles, but is not equivalent to, a cardioid.

See also

Circle-Circle Intersection, Circle Involute, Circular Sector, Circular Segment, Lens, Monty Hall Problem, Semicircle

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References

Hoffman, M. E. "The Bull and the Silo: An Application of Curvature." Amer. Math. Monthly 105, 55-58, 1998.Numberphile. "The Goat Problem." Dec. 24, 2022. https://www.youtube.com/watch?v=ZdQFN2XKeKI.Sloane, N. J. A. Sequence A133731 in "The On-Line Encyclopedia of Integer Sequences."Ullisch, I. "A Closed-Form Solution to the Geometric Goat Problem." Math. Intelligencer 42, 12-16, 2020. https://doi.org/10.1007/s00283-020-09966-0.Ullisch, I. "Correction to: A Closed-Form Solution to the Geometric Goat Problem." Math. Intelligencer 46, 1, 2024. https://doi.org/10.1007/s00283-023-10299-x.

Referenced on Wolfram|Alpha

Goat Problem

Cite this as:

Weisstein, Eric W. "Goat Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoatProblem.html

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