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 alleges possible large-scale fraud affecting hundreds or thousands of customers. Police appoint a technical-forensic consultancy team to analyze the mathematical evidence independently and issue the final opinion.
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
\[V(x)=x(L-2x)(W-2x), \qquad 0<x<\frac{\min(L,W)}{2}.\]The generator calculates the optimal box for common paper formats or a custom sheet. It then creates a full-size template, the advertisement, the complaint, the fictional patent, independent dossiers for the Consumer Association, cinema, and police consultancy, and the teacher-only solution.
Activity documents
Use the role packets to distribute the activity. In each tab, the red ZIP downloads every file in that section; the icons let you view or download each file. The template uses a PDF page matching the exact sheet dimensions: print at 100%, without “fit to page.”
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.
The advertising, complaint, and patent documents are fictional and for educational use only. They are not real advertising, legal advice, or a patent filing.