A Closed-Form Solution to the Geometric Goat Problem
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T
he geometric goat problem (not to be confused with the goat problem in probability theory) is a generic name for two different geometric problems in recreational mathematics: the interior goat problem and the exterior goat problem. Although only the former is dealt with in this paper, let us briefly review their respective historical origins for the sake of completeness. According to the article on the goat problem in Wikipedia,1 the exterior goat problem dates back to 1748, when it was first published in The Ladies’ Diary: or, Woman’s Almanack [23, p. 41]. This annual mathematical periodical featured, among other things, calendrical information, including times of sunrise and sunset and the phases of the moon, along with a list of mathematical questions the solutions to which were published in the subsequent issue. Reading the original text of the 1748 issue shows that in the course of history, a horse was transformed into a goat (see Figure 1): VIII. QUESTION 302. By Upnorensis. Observing a Horse tied to feed in a Gentleman’s Park, with one End of a Rope to his Fore-foot, and the other End to one of the Circular Iron-Rails, inclosing a Pond, the Circumference of which Rails being 160 Yards, equal to the Length of the Rope, what Quantity of Ground, at most, could the Horse feed?
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Besides the solution in the 1749 issue of the periodical [6, p. 25], a solution for the more general case in which the circular iron rails are replaced by a smooth convex curve was given in [9].2 According to [3], the interior problem was originally published in the American Mathematical Monthly’s first issue [7] in 1894, where the problem is posed as follows (see Figure 2): 30. Proposed by Charles E. Myers, Canton, Ohio. A circle containing one acre is cut by another whose centre is on the circumference of the given circle, and the area common to both is one-half acre. Find that radius of the cutting circle. More information on both problems can be found in [3], which also provides further references to variations of grazing goat problems; the original solution can be found in [17]. Nowadays, the interior case is the more prominent one. A possible reason is that some interesting generalizations of this problem have been investigated. For instance, instead of circles one can consider n-balls and ask for the radius of the cutting n-ball in an arbitrary dimension n. In [4, 14] it is shown that the radius of the cutting n-ball pffiffiffi converges to 2 as n tends to infinity. Another reason
Available online at https://en.wikipedia.org/wiki/Goat_problem. The author uses the term tethered-bull problem and does not refer to [23], but to a post in the Internet newsgroup sci.math.
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problem and then explain how to find a closed-form solution by means of elementary complex analysis.
A Closed-Form Solution
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Figure 1. The exterior goat problem. A goat is tethered to a fence surrounding a circular pond P (the white circle), where th
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