Box Folding Optimization Problem

Calculus 1 · Axiom Academy

EXAMPLE Box Folding Optimization Maximize the volume of an open-top box folded from a square sheet of cardboard. A piece of cardboard is 24 inches by 24 inches. We cut an equal square of side length x from each corner and fold up the four flaps to make an open-top box. What value of x maximizes the volume, and what is that maximum volume? Cut a square of side x from each corner (dashed red), then fold the four flaps up along the blue lines. The base of the finished box measures (24-2x) by (24-2x) , with height x . Nice work — you solved the classic box optimization problem end to end. Here is the playbook: Build the objective: express the quantity you want to optimize as a function of one variable — here V(x) = x(24-2x)^2 . Pin down the domain: the cut must be positive and smaller than half the side, so . Find critical points: solve V'(x)=0 , which gives x=4 and x=12 . Test and screen: x=12 is rejected (it sits on the domain edge and gives zero volume); confirms x=4 is a maximum. Answer: cut 4 -inch corner squares for a box with maximum volume 1024 cubic inches. Every optimization problem follows this same arc — the hard part is naming the objective function, so always start there.

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