Recreational mathematics planar boundary and area problem
Not to be confused with Monty Hall problem.
The goat grazing problem is either of two related problems in recreational mathematics involving a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing an exterior of a circular area. For the exterior problem, the constraint that the rope can not enter the circular area dictates that the grazing area forms an involute. If the goat were instead tethered to a post on the edge of a circular path of pavement that did not obstruct the goat (rather than a fence or a silo), the interior and exterior problem would be complements of a simple circular area.
The original problem was the exterior grazing problem and appeared in the 1748 edition of the English annual journal The Ladies' Diary: or, the Woman's Almanack, designated as Question CCCIII attributed to Upnorensis (an unknown historical figure), stated thus:
Observing a horse tied to feed in a gentlemen's park, with one end of a rope to his fore foot, and the other end to one of the circular iron rails, enclosing 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?
The related problem involving area in the interior of a circle without reference to barnyard animals first appeared in 1894 in the first edition of the renown journal American Mathematical Monthly. Attributed to Charles E. Myers, it was stated as:
A circle containing one acre is cut by another whose center is on the circumference of the given circle, and the area common to both is one-half acre. Find the radius of the cutting circle.
The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus. Both problems are intrinsically transcendental – they do not have closed-form analytical solutions in the Euclidean plane. The numerical answers must be obtained by an iterative approximation procedure. The goat problems do not yield any new mathematical insights; rather they are primarily exercises in how to artfully deconstruct problems in order to facilitate solution.
Three-dimensional analogues and planar boundary/area problems on other shapes, including the obvious rectangular barn and/or field, have been proposed and solved.[1] A generalized solution for any smooth convex curve like an ellipse, and even unclosed curves, has been formulated.[2]
^Bassett, Gilbert (2021-09-27). "The Goat in the City". The Mathematical Intelligencer. 44: 1–6. doi:10.1007/s00283-021-10120-7. ISSN 0343-6993. S2CID 244171722.
^Michael E. Hoffman, "The bull and the silo: an application of curvature," American Mathematical Monthly 105 (1998), 55–58
and 29 Related for: Goat grazing problem information
GrazingGoat Pictures is an Indian film and TV production house launched in 2011 by Hindi film actor Akshay Kumar and Ashvini Yardi. Its first film was...
The goat or domestic goat (Capra hircus) is a species of domesticated goat-antelope that is mostly kept as livestock. It was domesticated from the bezoar...
theorem Five circles theorem Gauss circle problem Gershgorin circle theorem Geometrography Goatgrazingproblem Hadamard three-circle theorem Hardy–Littlewood...
habitat, even replacing locally extinct wild goats. Feral goats are sometimes used for conservation grazing, to control the spread of undesirable scrub...
land-clearing purposes. Rent A Goat is part of a larger phenomenon called conservation grazing or targeted grazing whereby goats are used instead of traditional...
goat isn't a dairy goat, it's a billy goat." Besides losing its voice, the goat had other mechanical problems during the fair which may have been the...
The markhor (Capra falconeri) /ˈmɑːrkɔːr/ is a large wild Capra (goat) species native to South Asia and Central Asia, mainly within Pakistan, India, the...
sagebrush, making grazing sheep effective for conservation grazing. Research conducted in Imperial County, California compared lamb grazing with herbicides...
grazing. Rotational grazing can be used with ruminants such as cattle, sheep or goats; non-ruminants such as pigs can also be used. The herds graze one...
Feral goats are an invasive animal species in Australia. First arriving in the 18th century with European settlers, feral goat populations originated from...
Bhaji in Problem is a 2013 Indian Punjabi-language comedy film directed by Smeep Kang, who had earlier directed films like Carry on Jatta and Lucky Di...
likely brought the demise of the infant goat industry. Prior to the gold rush, flocks of grazing animals, goats and sheep were controlled by shepherds...
72 Miles, Bhaji in Problem, Fugly and Singh Is Bliing. After having a 20-year-long career in television, Yardi formed GrazingGoat Pictures with actor...
for use as animal fodder, either for large grazing animals raised as livestock, such as cattle, horses, goats, and sheep, or for smaller domesticated animals...
flat grassy top of the formation for grazinggoats, since few other species could scale the steep slopes. Goat Rock Beach polled as the second most popular...
Grazing pressure is defined as the number of grazing animals of a specified class (age, species, physiological status like pregnant) per unit weight of...
types of grazing animals, but frequent grazing over three to four years can ruin even established stands. Thus, kudzu only serves well as a grazing crop on...
bouquetin, was one of the four subspecies of the Iberian ibex or Iberian wild goat, a species endemic to the Pyrenees. Pyrenean ibex were most common in the...
from sheep to goat farming in order to meet export demands for cashmere wool has caused degradation of grazing lands. Compared to sheep, goats do more damage...
Organization, about 60% of the world's grassland is occupied by grazing systems. "Grazing systems supply about 9 percent of the world's production of beef...
(Capra pyrenaica), also known as the Spanish ibex, Spanish wild goat and Iberian wild goat, is a species of ibex endemic to the Iberian Peninsula. Four subspecies...
pasture. Rotational grazing is a variety of foraging in which herds or flocks are regularly and systematically moved to fresh, rested grazing areas (sometimes...
grazing animals, and wildlife. Open fields and grazing pastures with disturbed soils and open sunlight are often more susceptible. Protecting grazing...
the high price of fodder, grazing is the only choice. In the past, the Bedouins had developed a system for rotating grazing pressure in order to protect...
marine life. Grazing is blamed for much destruction. However, when grazing is modeled after nature, it can have the opposite effect. Cell grazing is a system...
Currently, grazing by goats and pigs is the best method for control of the vine. In Tennessee, a vegetation clearing company, that uses goats to control...
Global Information