Goat on a RopeDate: 8/6/96 at 9:43:2 From: Dave Nicholls, Information Analyst Subject: Goat Grazing from Edge of Circle Hi... I've got a simple question that I've never been able to work out. I'll set the scene: A circular field, diameter X, has a white picket fence around its edge. A goat's tether rope of length Y is attached to one of the these pickets. How long does the rope need to be so that the goat can graze exactly half the field? -Dave Date: 8/6/96 at 16:42:1 From: Doctor Anthony Subject: Re: Goat Grazing from Edge of Circle I shall use my own terminology for this problem, but here is the calculation: Since one cannot draw a decent diagram with ascii, I suggest you draw the following diagram on a piece of paper and refer to it 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 a starting value for x at about 0.7 radians 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/ Date: 05/25/97 From: Doctor Sarah You'll find more answers to interesting problems about tethering and grazing, some with illustrative diagrams, by searching the Dr. Math archives for the words "goat" and "cow" (just the word, not the quotes). Here are two more answers by Dr. Anthony: "Coordinate Geometry - Goat in a Circular Field": http://mathforum.org/dr.math/problems/booth8.22.96.html "The Goat in the Field Problem" http://mathforum.org/dr.math/problems/miller5.24.97.html -Doctor Sarah, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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