# Two cows are in a field…

In math club this past week, we didn’t have anything to work on explicitly. So we just made up a problem, based on a problem we encountered in the previous week.

Without further ado, here it is. You have a circular field, enclosed by a fence. Two cows Antonio and Barry graze in the field. They are each tethered to some place on the circle, tied with ropes of lengths $r_A$ and $r_B$ respectively.

The problem is: come up with a formula for the area of the region that both cows can graze together.

I love that we came up with the problem, and that we’re exploring it ourselves. It’s great that it’s so simply stated, and that it has a pretty tough solution. I love that it’s a generalization of something we did earlier. And I love that even this problem can be generalized further (e.g. we have $n$ cows).

What we did in 15 minutes:

We know we’re going to have a piecewise function of three variables. To start the problem, we make the circle a unit circle, we place Antonio at the point $(1,0)$ and we place Barry at $(\cos \theta, \sin \theta)$.

By the end of our math club meeting, we had one part of the piecewise function $f(r_A, r_B, \theta)$. We found where there would be no overlapping grazing area, where the function would be zero.

I have some sketches of the problem and the bit of solution we got together. I’ll put them below in a bit.