Evaluating Modeling Solutions: The Art Gallery Problem

Mathematical modeling plays an important role in the high school mathematics program in Australia. As part of the final examination to receive a high school diploma, students must successfully complete an extended modeling assignment. They then have a test on material similar to the modeling problem to assess their understanding of the problem.

One of the problems from the 1994 Specialist Mathematics examination is the basis for this quarter's Everybody's Problems column.The problem has five questions.The first three questions are fairly standard precalculus problems that require graphing calculators to find maximum and minimum values.They serve primarily to set up the student for the more interesting work on the final two questions.

It is the last two that are the most interesting and offer the greatest variety in solutions. On the Australian exams for which these questions were initially designed, techniques of calculus were the expected method of solution. However, the student solutions discussed in this article use only concepts from precalculus and geometry supplemented with graphing calculators. How would your students approach these problems?