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.


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