The Goat and/or Dog Problem

One particular problem type that appears periodically on the UNL Math Day PROBE I involves tying an animal to a building with a leash. The student then must make determinations regarding the area in which the animal can roam. Below are the problems that have appeared on the PROBE I since 1990.

1994 PROBE I #18
A square shack 30 feet by 30 feet is in the middle of a large field. A goat is tethered to one corner of the shack by a chain 60 feet long. She cannot get into, onto, or under the shack, but can graze anywhere else she can reach on her chain. What is the area of the portion of the field she can graze?


2001 PROBE I #2
A dog is tethered to the corner of the outside of a ten foot by ten foot building by a leash which is 14 feet long. How much area (outside of the building) can the dog roam?





2002 PROBE I #25
A dog is tied to the corner of a ten-foot by 20-foot shed on a rope having length 60 feet. Assume the dog starts out as pictured below and winds his way around the shed counterclockwise as far as he can go. What is the total area (in square feet) swept out by the rope?





With my students, I approach the three problems in similar ways. I want students to successfully obtain the correct solution. Next, I want them to write the area in which the animal can roam as a function of other features of the diagram. What matters here? The length of the leash? The dimensions of the building? The shape of the building? The direction of the rotation? Writing a function gives the student an opportunity to work towards solving for all possible cases and unearthing the features of the setting we can treat as invariant.

Geogebra provides a great medium for modeling this problem. A fun exercise involves asking what happens to the area in which the animal can roam as the length of the leash exceeds the perimeter of the figure. Modeling the overlap becomes a challenge quickly. Jerel Welker, a math coach in Lincoln Public Schools in Lincoln, NE, has used the most recent of the three listed problems as a wonderful professional development opportunity for middle school and high school math teachers.

What other features of these problems can we augment to challenge our students?

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