Optimisation: packaging design
ExampleThe packaging problem asks for the cylinder of least surface area that still holds 500 mL. I began by assuming the can is a right circular cylinder with negligible wall thickness, since the manufacturer specifies a 0.2 mm aluminium sheet. Letting r be the base radius in centimetres and h the height, the volume constraint gives h = 500/(πr²). Substituting into the surface area relationship produces a single-variable function that can be minimised.
It is important to note that optimisation plays a crucial role in a wide variety of real-world manufacturing contexts. Graphing the resulting function showed a clear minimum near r = 4.30 cm, which I then checked against the value my classmate obtained. The measured diameter of the sample can was 8.6 cm, which agrees with the model to within the tolerance of the ruler I used. However, I did not calculate how much that tolerance shifts the recommended radius, which limits how strongly I can state the conclusion.
