Optimising the Volume of Tubes
Calculus
A fixed amount of card can be rolled into an open tube many different ways — which gives the most volume? The answer is stranger than it looks.
Introduction
Take a rectangular sheet of card and roll it into an open-ended tube, with no base and no lid. A4 paper is about 620 cm²; use that or pick your own fixed area. For a fixed sheet area, is there a best radius that gives the most volume, or does the volume just keep climbing as you change the radius? Work through it with tables, a graph and a function before you decide.
Guiding Questions
- Roll a fixed sheet into a cylinder two different ways: rolling along the long edge, and rolling along the short edge. Which gives more volume? Calculate both.
- Fix the sheet area and write the tube's volume as a function of radius. Sketch the graph. Does it have a maximum — and if not, what does that tell you about your answer to question 1?
- Now add a base and a lid, cut from the same fixed area of card. Set up volume as a function of radius again. Does a maximum exist this time? Find it.
- Compare your optimum, with base and lid, to a real drink can's proportions. Why might the factory choose different dimensions?