# Goat tethered in a circular field

I need help on this question.

I was thinking of counting the area of the whole circle and then subtracting it with the area that is not eaten by the goat. But I don't know how to find this particular area. Was hoping the others can help me with this.

A goat is tethered at a point on the boundary of a circular grassy field, by a rope which is 1.5 times the length of the radius of the field. The field is surrounded by a fence which the goat cannot climb. What is the percentage of the grass in the field the goat can eat?

Let the radius of the field be r. Let the angle of ATB be 2β.

(T is the tethered point. With the rope at full stretch, the goat will be able to move in an arc from point A on the circumference to point B).

Many thanks!

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@Shobhit thanks man! How do you rate or vote or something in this thing? – Confusedwithmath Oct 10 '13 at 11:23
@Confusedwithmath you can vote the comment up by clicking the upside arrow that will show on the left of the comment. – SHOBHIT GAUTAM Oct 10 '13 at 19:05

A hint:

The circle is the unit circle, and the goat is tied to the point $(1,0)$. Denote the upper point where the stretched rope meets the fence by $A$. Connect $A$ with $(0,0)$, and let $A'$ be the point where the perpendicular from $A$ meets the $x$-axis. The upper half of the domain in question is then a sum/difference of various circular sectors and triangles whose areas can be computed from the given data.

Note that one of the sectors has central angle $\alpha$ with $\cos\alpha={3\over4}$, and the other sector has central angle $\beta=2\alpha$.

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The solution of the goat problem is here : http://mathworld.wolfram.com/GoatProblem.html

Even more diverting, the same problem in three dimensions and in n-dimensions space (the hypergoat in hyperspace) :http://fr.scribd.com/doc/14789360/Le-probleme-de-l-hyperchevre

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