Maximum Perimeter

At the right is the graph of the parabola

Consider the family of all rectangles with base along the x-axis inscribed underneath the parabola. Three members of this family are sketched in the diagram.

https://www.desmos.com/calculator/gws5larhbm

 The problem is to find the rectangle with the largest perimeter.

This looks like a calculus problem, except in grade 12 it is typically the area of the rectangle that is maximized. In replacing area with perimeter, we bring this problem into the grade 11 domain.

 The first question for the students is to find a good parameter to label the different rectangles. Then the perimeter formula turns out to be a quadratic function of that label. So now the students have two different but closely related parabolas to work with. Interesting.

 We’d like this to be a grade 10 problem––in our new vision of grade 10, students should be working more with the quadratic formula. But currently in Ontario, this problem fits the grade 11 curriculum.