(also found here.)
Graphing calculator pictures:
A picture of the maximum volume
A picture of one non-extraneous value of x where the volume = 400.
I think this problem tied together many things:
- Why domain of a function is important in the real world
- How to adjust windows on the graphing calculator
- How to find the relative maximum on a graphing calculator
- Why the relative minimum of the graph didn't matter (out of the domain.)
- What an extraneous root looks like (the graphing calculator gave a third answer outside of the domain for a volume of 400 cubic cm.)
- How to work in groups as a team (though each student did have to create their own boxes)
- How letter e) could have been solved two ways: by using the so far unknown intersect feature of the graphing calculator or by setting the equation equal to 400, subtracting 400 and finding zeros...this was cool for kids to see!
- The eventual tie-in to polynomial graphs
- The actual making of the boxes
Perhaps my favorite comment (several mentioned that they loved doing this) was when a student said that when he heard we were doing an investigation today, he was nervous, but then during and afterward, he was excited and really enjoyed himself.
I was observed during this class, and in the write-up, my department chair noted that every student was focused, on-task, and working together for the entire period. He wrote that students exclaimed: WE GOT IT!!!! and that "the teacher smiles and laughs with glee." Ha. Yep. Every damn day.