Being as people were bored and someone asked for a puzzle....

[Project Monkey]

The chain is inflexible? It's never going to work then.

Depends. Which part of the goat is attached to the inflexible chain, how big is the goat, how long is its neck etc. If it's a very small field that I supposed it's feasible.

[chef]

Wibble

[quackhandle]

Simple, it's 42.

qh

[greenlake]

[ASB]

So, circle 1 in the field.

Given the mythical goat is tether at the inside edge then the length of the tether defines another circle.

However it is bounded by the first circle, so defines an arc.

So the question is

"What is radius of a circle which is centered on the edge of another circle where the overlapping arc is equal to half the entire area of the overlapped circle"

At least I think that is the question.

[xoggoth]

0.7071 of radius?

[ASB]

That was my initial thought. But having drawn it I couldn't be arrised to try and figure it out for sure.

[Doggy Styles]

0.7071 of radius?

If it was centred so the goat's circle was entirely within the field.

But it's centred in the edge of the field so you are looking at the area of overlap of the two circles.

[Contreras]

So, circle 1 in the field.

Given the mythical goat is tether at the inside edge then the length of the tether defines another circle.

However it is bounded by the first circle, so defines an arc.

So the question is

"What is radius of a circle which is centered on the edge of another circle where the overlapping arc is equal to half the entire area of the overlapped circle"

At least I think that is the question.

Yes, that is the question. Or is it?

Tether needs to be > radius of the field otherwise clearly it would reach less than half the field area. It needs to be < √2 or clearly the goat would reach more than half the field. So somewhere between 1.0 and 1.4ish.

However the maths seems so impossibly difficult that it must be a trick question. Is the tether fixed to a single point? I'm not sure that was stated. Anyway, off to google now...

[EternalOptimist]

The answer is to give the goat a tether that will allow it to reach the far side of the circle.
i.e. = D

This obviously means that the goat can cover the whole field and therefore eat twice the amount specified in the opening question.

The resolution is to introduce a second goat on the other side with a tether that = D.

Each goat will scoff half the grass