Calculus Cases: The Perfect Popcorn Box
A cinema chain is promoting its new container with a bold statement: “The greatest-capacity box: more popcorn from the same cardstock.” A consumer association says it has built, from the same sheet, a box that holds more popcorn, and alleges possible large-scale fraud. The Chief Inspector, a role the teacher takes on, divides the case among three teams: the Consumer Association builds boxes with different cuts and makes the allegation; the cinema team builds the box registered in the patent; and the police technical-forensic consultancy works out on its own, without seeing the patent, which box is the largest, and issues the opinion.
The case follows the format of every Calculus Case, with one difference: the boxes are built in class, because it is cheap and it is the most enjoyable part.
- Launch, at the end of a class (15–20 min). You project the presentation: the advertisement, the complaint, the question and the three teams of your class. Each pair or group receives its packet, and each Association pair chooses its cut.
- At home. The consultancy models, differentiates and proves the maximum without knowing the patent (45–60 min); the Association works out the volume its cut predicts and the cinema prepares the construction (15–20 min).
- Closing class (60 min). The boxes are built and measured (phase 1, 25 min); the consultancy receives the complaint and the patent and, with the popcorn, the transfer test is run (phase 2, 15 min); the consultancy issues its opinion (12 min) and every student does an individual check (5 min).
The box is made from a rectangular sheet with sides (L) and (W). A square with side (x) is removed from each corner—always the same size—and the four sides are folded up. The height is (x), while the base measures ((L-2x)\times(W-2x)), so
\[V(x)=x(L-2x)(W-2x), \qquad 0<x<\frac{\min(L,W)}{2}.\]Materials generator
The configurator guides you in four steps. The only essential setting is the number of students: from it, the website forms the three teams and works out how many copies of each document to print and how many sheets and how much popcorn you need. Everything else already has a recommended value.
Activity documents
Start with the printing and distribution guide: it says what to print, how, how many copies and when each document is handed out. The teacher packet contains the solution and must not circulate. In each tab, the red row downloads a ZIP with all its content; the icons let you view or download each file.
Why the generated cut is optimal
Differentiating the volume function gives
\[V'(x)=12x^2-4(L+W)x+LW.\]Its two roots are
\[x=\frac{(L+W)\pm\sqrt{L^2-LW+W^2}}{6}.\]The root with the minus sign is the only one in the physical domain:
\[x^*=\frac{(L+W)-\sqrt{L^2-LW+W^2}}{6}.\]The generator compares this critical point with the interval endpoints, where volume is zero, and also checks that (V’‘(x^)<0). Therefore, (x^) produces the global maximum within the model: a fixed rectangular sheet, four equal square cut-outs, an open box, and no added material. Near the optimum the volume curve is almost flat, which is why popcorn can hardly tell a good box from the best one: what settles the case is the derivative.
The advertising, complaint, and patent documents are fictional and for educational use only. They are not real advertising, legal advice, or a patent filing.