Solving David's Puzzling Circular Pasture Question

[Dabonez]

My maths teacher recently gave me a weird question which really got my attention, but I have not managed to solve it so far. It can't exactly be described as homework, although it would give me a huge bonus if I managed to hand in the results.


There is a pasture in the shape of a circle. A cow is tied to a point on the circle. How long does the rope have to be, so the cow manages to pasture exactly 1/2 of the grassland?


I'm looking forward to everybody's opinions, and thanks in advance!


David

 

[zgozvrm]

To start, assume the pasture has radius R, then the area of the pasture is \pi R^2.

It may help to assume that the center of the pasture lies on the positive x-axis, and that edge of the pasture passes through the origin.

Now, assume that the cow's rope has radius C and is tied to the origin. This is the center of a second circle having area \pi C^2.

You are looking for the intersection of these two circles.

Note that you can assume C>R if it is to cover half of the pasture.

 

[zgozvrm]

Also, you know that C < R\sqrt{2}, since having C = R\sqrt{2} would mean that circle C (the cow's circle) would intersect circle R (the pasture circle) at (R,R) and (R,-R). This intersection is obviously greater than 1/2 the area of circle R.

So now we have R < C < R\sqrt{2}.

 

[zgozvrm]

Assume the intersecting points between the two circles are A (at the top) and B (at the bottom). Line segment AB is a chord of both circles, and thus we have 2 circle segments.

This should give you enough information to get you on the right track...