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 and 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 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 and we place Barry at .
By the end of our math club meeting, we had one part of the piecewise function . 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.