The Box Problem

One square sheet of card, $20$ cm on a side. Cut equal squares from the four corners, fold up the flaps, tape the seams: an open-top box. Everyone in the room gets the same sheet. Whose box holds the most?

The task

Build first. Choose a cut size, make the box, and compute its volume in ${\text{cm}}^3$. Post the class data — cut size against volume — and stare at it: the numbers rise, peak, and fall. Now the real work: could any cut beat the best box built in this room? Write the volume as a function of the cut $x$, decide which values of $x$ even make sense, and use this unit’s tools to find the cut nobody can beat — then prove nobody can. Then push: does your winning cut survive a bigger sheet — $30$ cm, $40$ cm? Conjecture the rule for a square sheet $s$ centimetres on a side and defend it. And what happens to all that tidiness if the sheet is $20$ cm by $30$ cm instead?

Facilitation notes — for the teacher

The physical boxes matter: the class scatter plot never quite matches the model — tape, flap thickness, ragged cuts — and naming that gap honestly is better mathematics than hiding it. Expect a fight over the domain; a cut of $10$ cm leaves no box at all, and the group that says so out loud has found the constraint. The proof question separates “the biggest we found” from “the biggest there is” — the sign of the derivative on each side settles it. Fast groups: the square sheet’s winning cut is always the same fraction of the side — let them find the fraction — and the $20\times 30$ sheet resists tidy numbers on purpose.

What mathematics tends to surface

The whole optimization playbook, in order and by necessity: a constraint that turns two variables into one, a domain the story itself imposes, a derivative whose zero locates the peak, and a sign argument that turns “best so far” into “best, full stop”. The class data even supplies the misfit between model and cardboard. Optimization names the playbook right after.

Where it leads

The Packaging Brief runs today’s playbook on a package that actually exists, with a client who wants the honest accounting. The optimization problems that follow — cost, revenue, distance — change the story but never the plays.

The answer is not on this page

The winning cut, the proof, and the general rule all happen at the boards — bring scissors.

Curriculum connection

B2.4

solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations Sample problem: The number of bus riders from the suburbs to downtown per day is represented by $1200(1.15{)}^{-x}$, where $x$ is the fare in dollars. What fare will maximize the total revenue?

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B2.5

solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results Sample problem: A bird is foraging for berries. If it stays too long in any one patch it will be spending valuable foraging time looking for the hidden berries, but when it leaves it will have to spend time finding another patch. A model for the net amount of food energy in joules the bird gets if it spends $t$ minutes in a patch is $E=\frac{3000t}{t+4}$. Suppose the bird takes 2 min on average to find each new patch, and spends negligible energy doing so. How long should the bird spend in a patch to maximize its average rate of energy gain over the time spent flying to a patch and foraging in it? Use and compare numeric, graphical, and algebraic strategies to solve this problem.

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