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 second 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 10 domain. We think that is a good problem because right way the students are faced with the task of find a good parameter to work with.

The formula for the perimeters of the different rectangles turns out to be a quadratic function, 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 but currently it fits the grade 11 curriculum.