TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Prerequisites

Optimization strategies may be useful, point of tangency to a curve, and system of nonlinear equations.

Time allotment - time management

This could be a big problem and may need some guidance to see how to determine the necessary conditions for a maximum area for the inscribed circle, i.e. the geometric constraints. We suggest permitting 15 minutes for brainstroming, picture drawing, and beginning the initial strategy.

If the teacher wishes to suggest the approach, we offer in the Possible Solutions section hints that can be given to move the class along. Setting up of the constraints together with the complexities of solving (if possible) need time.

It may just be that articulating strategies, going after a few plausible candidates, and offering up some plots will be sufficient for the purposes of understanding what the problem is asking for. These plots may motivate the multiple tangency approach we use in our Possible Solutions.

Expectations

We would expect students to sketch quite a bit before making necessary mathematical conjectures.

Future payoffs

Students will have utilized a number of different calculus skills in doing this problem and reviewed slopes of lines and orthogonal lines.

Extensions

One can consider other functions which give rise to different geometric configurations. Some quite regular and others quite pathological, e.g., consider transcendental functions.

One context may be that we have purchased a good bit of scrap metal sheets (dimension in inches say) in the shape of the region described and we wish to salvage what we can. So we seek to cut out the largest circular disk from the material to make platters.

References and Sources