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6.AH. TETHERED GOAT
A goat is grazing in a circular field and is tethered to a post on the edge. He can reach half of the field. How long is the rope? There are numerous variations obtained by modifying the shape of the field or having buildings within it. In recent years, there has been study of the form where the goat is tethered to a point on a circular silo in a large field -- how much area can he graze?
Upnorensis, proposer; Mr. Heath, solver. Ladies Diary, 1748-49 = T. Leybourn, II: 6-7, quest. 302. [I have a reference to p. 41 of the Ladies' Diary.] Circular pond enclosed by a circular railing of circumference 160 yards. Horse is tethered to a post of the railing by a rope 160 yards long. How much area can he graze?

Dudeney. Problem 67: Two rural puzzles -- No. 67: One acre and a cow. Tit Bits 33 (5 Feb & 5 Mar 1898) 355 & 432. Circular field opening onto a small rectangular paddock with cow tethered to the gate post so that she can graze over one acre. By skilful choice of sizes, he avoids the usual transcendental equation.

Arc. [R. A. Archibald]. Involutes of a circle and a pasturage problem. AMM 28 (1921) 328 329. Cites Ladies Diary and it appears that it deals with a horse outside a circle.

J. Pedoe. Note 1477: An old problem. MG 24 (No. 261) (Oct 1940) 286-287. Finds the relevant area by integrating in polar coordinates centred on the post.

A. J. Booth. Note 1561: On Note 1477. MG 25 (No. 267) (Dec 1941) 309 310. Goat tethered to a point on the perimeter of a circle which can graze over ½, ⅓, ¼ of the area.

Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968.

No. 8: "Don't fence me in", pp. 87. Equilateral triangular field of area 120. Three goats tethered to the corners with ropes of length equal to the altitude. Consider an area where n goats graze as contributing 1/n to each goat. What area does each goat graze over?

No. 53: Around the silo, pp. 71 & 112-113. Goat tethered to the outside of a silo of diameter 20 by a rope of length 10π, i.e. he can just get to the other side of the silo. How big an area can he graze? The curve is a semicircle together with two involutes of a circle, so the solution uses some calculus.

Marshall Fraser. A tale of two goats. MM 55 (1982) 221 227. Gives examples back to 1894.

Marshall Fraser. Letter: More, old goats. MM 56 (1983) 123. Cites Arc[hibald].

Bull, 1998, below, says this problem has been discussed by the Internet newsgroup sci.math some years previously.

Michael E. Hoffman. The bull and the silo: An application of curvature. AMM 105:1 (Jan 1998) ??NYS -- cited by Bull. Bull is tethered by a rope of length L to a circular silo of radius R. If L  πR, then the grazeable area is L³/3R + πL²/2. This paper considers the problem for general shapes.

John Bull. The bull and the silo. M500 163 (Aug 1998) 1-3. Improves Hoffman's solution for the circular silo by avoiding polar coordinates and using a more appropriate variable, namely the angle between the taut rope and the axis of symmetry.

Keith Drever. Solution 186.5 -- Horse. M550 188 (Oct 2002) 12. A horse is tethered to a point on the perimeter of a circular field of radius 1. He can graze over all but 1/π of the area. How long is the rope? This turns out to make the problem almost trivial -- the rope is 2 long and the angle subtended at the tether is π/2.