A farmer has a field 100 (metres, feet, yards, it does''t matter) in diameter. He also owns a goat and a rope. He ties the rope to a fence post at the edge of the field, and ties the goat to the other end of the rope, such that the goat can eat exactly half of the grass.
How long is the rope?
Originally posted by CauselessOnepi*R^2 = 2*pi*r^2
A farmer has a field 100 (metres, feet, yards, it does''t matter) in diameter. He also owns a goat and a rope. He ties the rope to a fence post at the edge of the field, and ties the goat to the other end of the rope, such that the goat can eat exactly half of the grass.
How long is the rope?
R = 100
solve for r.
Oops. I assumed the goat's post was in the center. Never mind.
Here's a start then:
http://mathworld.wolfram.com/Circle-CircleIntersection.html
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Over 50, but less than 100. After lots of writing, figuring, and scribbling, the only way I can see to solve this is to focus on the fact that units don't matter. Therefore my other answer is that the rope should be 10 smomofos long, and it is up to someone else to determine the conversion factor from whatever units the diameter of the circle is measured in and the length of 1 smomofo.
Originally posted by ark13Which corner of the circular field will you use?
The field is 100*100=10,000. The goat must eat 5,000 of it. Tied to a post at a corner, the goat's possible roaming area inside the field will be a quarter circle. Therefore,
5000=.25*pi*r^2
20,000=pi*r^2
6366.2=r^2
79.79=r
Originally posted by CauselessOne35.35 units
A farmer has a field 100 (metres, feet, yards, it does''t matter) in diameter. He also owns a goat and a rope. He ties the rope to a fence post at the edge of the field, and ties the goat to the other end of the rope, such that the goat can eat exactly half of the grass.
How long is the rope?
nevermind never read the question right
GV
i have 3 answers to this problem....
1) the ropp is as long as it has to be and is not to long and not to short but lets the goat eat half of the filed
2) 50 meters
30 in order to work out the answer you need to draw the feild and then slect a piont on on of the edges where the rop has been tyed. then positsion a half squrle that has the area of 50% of the feild. then mesure the distance from the goat to the piont where the rope is tyed to the fence
Originally posted by CauselessOneUsing the formula from the page AThousandYoung posted. Substitute d=50, R=50, A=1250*pi and solve for r. Of course actually solving for r is pretty much impossible (Maple won't do it even) so the easiest way is a Newton's Method.
Somewhere around there IIRC, but how did yoy calculate it?
BTW, the only way I know is to draw the damned thing!
Originally posted by XanthosNZthere is no solution thats why it won't solve it, the determinant < 0
Using the formula from the page AThousandYoung posted. Substitute d=50, R=50, A=1250*pi and solve for r. Of course actually solving for r is pretty much impossible (Maple won't do it even) so the easiest way is a Newton's Method.
.5 sqrt (r^2(100-r) (100+r) = 1250pi
r^2 (100^2-r^2) = (2500pi)^2 ~ 625*10^5
0 = r^4 - 10^4 * r^2 + 625 * 10^5
r^2 = (+100^4 +/- SQRT(10^8 - 4 * 625* 10^5)) /2
10^8 - 4* 625* 10^5 =10^8 - 2.5 * 10^8 < 0
i think we violated some limiting condition of the problem at mathworld.