Tethering Goats

I have been thinking about what it means to “do math” and I’ve decided that it is more than solving equations/problems. That’s finding answers, and math is really so much more than that. We had a professional development meeting the other day, and an expert in math literacy spoke to the department. We have been focusing on literacy quite a bit the last few years at my school, but not specifically what that means in a math class. One of the things that struck me after the recent meeting was that when we read, we construct pictures in our minds about what is happening. When we read narratives it is okay if those pictures don’t match the words exactly, as long as we have the action straight. When reading mathematics, on the other hand, the need for those pictures to be exact and precise, and in many cases animated, is very important.

So I have a feeling that doing math means constructing models, both physical and abstract. It’s not about the questions and answers, but about the pictures we make in our heads. Once we have a good picture, there are lots of questions for which we can find answers.

For example, there is a classic problem in which we consider tethering a goat with a rope of a particular length, staked to the ground in some position. The picture in your head probably resembles the image here, and you can even animate it, imagining the goat wandering around eating whatever grass is within reach.

As a mathematician, I start wondering about how much grass is available for the goat to eat? How far can the goat reach? I’m sure you can figure out that the goat can reach all the grass within a circle centered at the stake in the ground. The radius of the circle is the length of the tether. How do I know that this is a circle? Because one definition of a circle is the set of points equidistant from a given point. The given point is the stake, and all the points at the farthest reach of the goat are the length of the tether (and the distance the goat can stretch its neck) from the stake. All of a sudden, I can create an abstract model in my head – I no longer need the picture of the goat to think about where it can graze. My picture is just a point (the stake) and the circle showing the limit of the goat’s range. If I want, I can include a radius (the tether) in the model as well. I can also animate my model, picturing the radius sweeping around the circle and the goat tests the boundary all the way around.

The other aspect of an abstract model is that I am not interested in the actual length of the tether, so I use a variable to represent that length; I’ll use t. Now, it does not matter whether the goat is on a tether that is 5 feet long, or 18 feet long. I can even consider a tether that is a mile long, if I assume the field is really big and has nothing in the way that the tether could get wrapped around. (I’ll think about those possibilities later.)

Why is the model important? Because it allows me to answer lots and lots of questions about the situation! Here are a few:
1. If the tether is 8 feet long, what is the area in which the goat can graze?
2. Suppose we want to fence in the grazing area, just in case the goat chews through the tether. How many feet of fence would we need if the tether were 10 feet long?
3. If a goat needs 50 square feet of grazing space each day, how long should I make the tether?

Since I have the model of a circle in mind, I can answer all these questions by using what I know about circles: The area within a circle is equal to pi times the square of the radius, and the perimeter or circumference of the circle is equal to pi times twice the radius, or pi times the diameter.

Here’s another question: Suppose the tether were 10 feet long, and that the goat wanders randomly within its grazing area. What percentage of the time is the goat within five feet of the stake? For this question, I need to modify my model slightly. I am no longer thinking of just the circle with a radius, but two circles, one inside the other, and a point (the goat) wandering around inside. I am making a big assumption here: the goat wanders randomly. A particular goat may be very obstinate about this, and decide to sit in one spot all day; another goat may decide to constantly try the limits of the tether, and remain at the outer circle all day. As a mathematician, however, I am not interested in these special cases, but in the general wandering goat. Anyway, my model now has a circle with radius five inside a circle of radius 10, and a point moving around inside. The probability that the goat is within the inner circle is the ratio of the area of the inner circle (the part I’m interested in) to the area of the outer circle (the total allowable wandering space). In other words: (5^2*pi)/(10^2*pi). Squaring, we get (25pi)/(100pi), and simplifying, we get ¼ or 25%.

Now, being a mathematician, this is not where the problem ends. Answer getting is not the purpose. Completely understanding the model is what this is all about. So, I start generalizing a little. Remember that I was going to use a variable for the length of the tether? So the question now becomes “what percent of the time does the goat spend within a region halfway to the edge of his grazing area?” Using the same formula as above, but using t for the length of the tether, and ½t for half that length, we have:
(½t)^2*pi / t^2*pi
= ¼ t^2 / t^2
= ¼ or 25%
In other words, no matter how long the tether, the area of the inner circle is always 25% of the area of the outer circle, if the radius of the inner circle is ½ the radius of the outer circle. I keep jumping between the physical model of the goat and the abstract model of the concentric circles.

As I write this, I am thinking about my model, and I realize that if the inner circle is ¼ the area of the outer circle, then I can divide the circle into quarters differently than I usually do. I bet you think about dividing the circle into quarters like the diagram on the left. But the diagram on the right also has the circle divided into four equal parts! Again, I start animating the pictures. The left diagram starts rotating around, and folding along the lines, and it's pretty clear to me that all four quadrants have the same area. The diagram on the right is not so easy. I can rotate the picture and prove to myself that the three sections in the outer ring are all equivalent, but how can I manipulate the model to make that center circle, which I know has the same area, match up with the other three sections? This will take some thought...

Here are some variations on the inital situation that also require some changes to the model. (I'm going to have to explore these in a later post.) What if the tether were staked along the side of a building? What if the stake was at the corner of a building? What if the tether was long enough so that the goat could walk all the way around the building?

I am beginning to feel like a goat on a really long tether, exploring the outer reaches of my model's boundaries!

By the way, here are the answers to the questions above: (I apologize for the format; I am still learning the capabilities of posting pictures and math symbols here.)
1. 64*pi which is about 201.062 square feet of grass
2. 20*pi which is about 62.832 feet of fence
3. sqrt(50/pi) or approximately 3.989 foot long tether