This is another post about sessions I attended last weekend at the Asilomar Northern California CMC conference. (To read the whole set, start here.)
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: