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Goat on a Rope Problem

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A zero dimensional goat is attached by a rope to a point on the perimeter of a circular field (two dimensional).

How long should the rope be (in terms of the radius of the field) so the goat can reach and graze exactly half of the grass of the field?

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Solution Summary

The attached 2-page Word document contains a diagram along with many lines of worded reasoning and calculations which take this problem as far as elementary expressions can go (no concrete numerical solution).

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Suppose the radius of the circular field is . The following graph shows how the goat can reach and graze half of the field. In this graph, the solid circle is the circular field and the broken circle is the area the goat can reach. This broken circle is centered on a point of the perimeter of the circular field and has a radius . The shaded area is the intersection of the two circles and is ...

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