Minimum-Cost Packaging
Calculus
A box needs a fixed volume, but the material for the lid and base costs more than the material for the sides. Work out the dimensions that minimise total cost, not just total surface area.
Introduction
Standard optimisation problems minimise surface area for a fixed volume. Real packaging is different: corrugated card for the base often costs more per square centimetre than the side panels, because it needs to bear weight. This idea asks what happens to the optimum dimensions once you price the faces differently instead of treating every square centimetre as equal.
Guiding Questions
- Pick a fixed volume for an open or closed box. Give the base and the sides different costs per unit area, and set up a cost function in terms of one variable.
- Use calculus to find the dimensions that minimise cost, and check your stationary point is a minimum.
- How do the optimum dimensions shift as the base becomes relatively more expensive? Try a few cost ratios and describe the pattern.
- Compare your cheapest box with a real shipping box you can measure. Does the manufacturer seem to be minimising cost, or something else?