The Soda Can Optimization Problem: Getting Close to the Real Thing

Abstract

Optimizing the dimensions of a soda can is a classic problem that is frequently posed to freshman calculus students. However, if we only minimize the surface area subject to a fixed volume, the result is a can with a square edge-on profile, and this differs significantly from actual cans. By considering a more realistic model for the can that consists of six components, including varying wall thickness, and minimizing the total metal volume, we arrive at an optimal can that has dimensions more in-line with actual manufactured cans. This model indicates an optimal radius of 2.83 cm and an overall height of 15.2 cm, which is closer to the dimensions of real cans today than what is obtained from assuming the can is a right circular cylinder. The calculations involved would serve as a useful undergraduate modelling project.

Keywords:

• optimization
• calculus
• modelling

Additional information

Notes on contributors

Kirthi Premadasa

Kirthi Premadasa obtained his MS and Ph.D. from Purdue University and has over 22 years of college teaching experience in Sri Lanka and the US. He is an Associate Professor of Mathematics at the University of Wisconsin-Baraboo/Sauk County. He is a Wisconsin Teaching fellow and was awarded the UW Colleges Chancellor award for teaching excellence in 2012 and the UW System’s Alliant Energy Underkofler Excellence in Teaching Award in 2013. He is also the vice-chair of the UW-Colleges Math Department

Paul Martin

Paul Martin obtained his Ph.D. in Mathematics at University of Wisconsin in 1994. Since then he has been teaching at the UW-Colleges. He is a full Professor of Mathematics and Associate Chair of the UW-MC mathematics department. He has won several teaching awards and he strives to make mathematics real to his students through applications and history.

Bryce Sprecher

Bryce Sprecher is a student at the University of Wisconsin-Baraboo/Sauk County. He plans to do a double major in computer science and biology with a neuroscience focus. He also plans to go to graduate school and research /work in artificial intelligence, nanotechnology, nano robotics, cognitive modeling, and neural prosthetics.

Lai Yang

Lai Yang was born in Shanghai, China in 1993. She came to Wisconsin after her freshman year at East China Normal University. Then she decided to study at University of Wisconsin Baraboo/Sauk County. She plans to study Actuarial Science.

Noah-Helen Dodge

Noah-Helen Dodge is a Mathematics major at UW-Baraboo/Sauk Co. She hopes to someday earn her Ph.D. in mathematics and become a professor.