bonanova 83 Report post Posted March 8, 2013 A goat is tethered to a circular silo of radius r = 29 yd that is centered in a circular grassy field of radius R. The goat is just able to keep the silo wall clear of grass. He is also able to eat half of the grass in the annular field. How big must that field be? Quote Share this post Link to post Share on other sites
0 Yoruichi-san 116 Report post Posted March 9, 2013 (edited) ...actually make that 120, found a math error *whistling*...and I meant "semi-circle of radius 29*pi in that last part... Eh, the integral is not that bad...only one application of parts is needed, and a fairly simple one at that Edited March 9, 2013 by Yoruichi-san Quote Share this post Link to post Share on other sites
0 Rob_Gandy 2 Report post Posted March 9, 2013 R ~ 254.84 or sqrt{2([58Pi]^{2}-27^{2})} Quote Share this post Link to post Share on other sites
0 bonanova 83 Report post Posted March 9, 2013 R ~ 254.84 or sqrt{2([58Pi]^{2}-27^{2})} Smaller than the silo footprint. To clarify: Quote Share this post Link to post Share on other sites
0 Prime 15 Report post Posted March 9, 2013 (edited) The goat in the picture is about 50 yards long and 40 yards tall. I would not like to meet it in a dark alley. For any rope length, a field could be built, such that the goat would have access to half of it. I guess, we are looking for the minimum size field. The minimum rope should allow the goat to reach around half of the silo. Thus the rope length is 29Ï€ yards. After that, it seems, the problem requires solving an integral. Edited March 9, 2013 by Prime Quote Share this post Link to post Share on other sites
0 Yoruichi-san 116 Report post Posted March 9, 2013 R~127 Although it's entirely possible I made a math error somewhere *shrugs*. I'm too sick/lazy to make a pretty diagram, but I divided the area the goat spans in half along the line of symmetry. The 'funky' part (i.e. where the leash is at least partially along the silo wall) can be described by the function r(theta)=29(pi-2*theta+2sin(theta)). To get the area, integrate 1/2*r(theta)^2 over theta from 0 to pi/2. Double that and subtract the area of the silo. The part where the goat's leash does not touch the silo is a semi-circle of radius 29. The rest is a little geometry, a little algebra. Quote Share this post Link to post Share on other sites
0 bonanova 83 Report post Posted March 9, 2013 I am wondering whether my analysis is off a bit. I get a slightly smaller R than Y-San does. But RG's answer gives exactly twice hers, meaning he may have given diameter instead of radius. So there is the possibility they are both correct (with a misplaced factor 2 in RG's case - a typo more or less.) Anyone else want to give it a shot, I'll go back over my equations meantime. Nice going, both, in any event. Prime is correct, it involves an a messy integral. And to be clear, the area the goat covers is half the annular field of grass. Quote Share this post Link to post Share on other sites
0 bonanova 83 Report post Posted March 9, 2013 Nice going. Quote Share this post Link to post Share on other sites
0 dark_magician_92 4 Report post Posted March 9, 2013 If i got the question right, then R is approx. 125.54 Quote Share this post Link to post Share on other sites
0 bonanova 83 Report post Posted March 10, 2013 If i got the question right, then R is approx. 125.54 dm92 your answer answer agrees with that of several others. Y-san and I get a slightly smaller number. It puzzles me (npi) what is the difference in solving this that leads to two different solutions. Maybe the wording of the OP is ambiguous. R = 121.14 yd Quote Share this post Link to post Share on other sites
0 Tom Laidlaw 0 Report post Posted September 18, 2013 121.14 seems to be correct. The area eaten by the goat is (29pi)^3/(3*29) +(29pi)^2+pi/2 = 21730.2. The field is x^2*pi=2(21730)+29^2*pi (no grass in the silo area so the radius is a little bigger) Quote Share this post Link to post Share on other sites
A goat is tethered to a circular silo of radius r = 29 yd that is centered in a circular grassy field of radius R. The goat is just able to keep the silo wall clear of grass. He is also able to eat half of the grass in the annular field.
How big must that field be?
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