10217

# Maximizing the Volume of a Cup Made from a Square Sheet of Paper

This Demonstration studies the maximization of the volume of a cup shaped like a truncated square pyramid made from a square sheet of paper of size 12×12 cm. This may be a new kind of problem in Origami.

### DETAILS

The volume of this 3D figure is . We look for the maximum value of .
By and we have
and .
From these two equations we get and .
Using Mathematica to factor, .
Again using Mathematica to solve, it is easy to see that we get the maximum value when and , and the maximum value is .
According to the referee of the Journal Origami Society of Japan, this is a new problem proposed by the authors.
The origami wrapping problem, which has been studied by some mathematicians, looks for the biggest object that can be wrapped with a sheet of paper [1].
Reference
[1] M. L. Catalano-Johnson and D. Loeb, "Problem 10716: A Cubical Gift," American Mathematical Monthly, 108(1), 2001 pp. 81–82.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Curriculum Standards

US Common Core State Standards, Mathematics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.