Kevin Rees presented two variations on a classic volume optimization problem. In the traditional problem, you start with a square piece of cardboard, cut off congruent squares at the four corners, and fold up the sides to make a box. Apparently this has been a standard application in calculus textbooks going back perhaps 150 years! I first encountered the problem in the Chakerian-Stein-Crabill Algebra 1 book, which is now out of print. Of course, in Algebra 1 you do not use calculus to maximize the volume, but it is a good way to introduce the power of multiple representations of functions.
Kevin teamed up with a colleague, and they came up with six variations on this problem for their calculus class. His school has dropped APs, which means there is more time to go in depth and pursue projects such as this one. Teams of students were each assigned one of those problems, and spent some time actually making models using paper, before tackling the calculations. (This turned out to be a crucial step, as we learned from our own experience working on this!)
One of the problems Kevin showed us was this one:
You cut off these kites at the four corner, and fold up the resulting trapezoids until they meet, making an ashtray shape. (Technically: a frustum of a right square pyramid.) The dimensions of the inner and outer squares are fixed. How far from the corner should the cut be to maximize the volume of the final polyhedron?
The algebra / trig / calculus manipulations turned out to be very complicated, and thus satisfying if one managed to get to the endpoint. While my team worked on this, I confess I got sidetracked into trying to solve the problem experimentally using Cabri 3D. This was big fun, though I didn't finish until the next day, at home.
Here is the figure where the polyhedron has maximum volume (or at any rate, close to the maximum.)
And here is some of the work that went into making that construction:
Do you think my preference for a geometric / engineer's solution as opposed to an algebraic / mathematical solution is cheating? Well, you are entitled to your opinions -- and I have a right to have fun! Thanks to Kevin for giving me an interesting project.
(As it turns out, I contributed some Cabri 3D figures to a different session, that time in order to get to a correct mathematical proof -- read about that in my next post!)