10753

# The Mug That Holds the Most Liquid

An important goal in engineering is to reach maximum function with minimum resources. This Demonstration answers the question: how do you design a cylindrical mug to contain the maximum liquid for a given volume of material?

### DETAILS

Snapshot 1: the objective function of liquid volume, which is the volume of a cylinder , where is the radius and is height
Snapshot 2: the constrained function of the mug volume, which is the sum of the outer shell and the bottom
Snapshot 3: the optimal solution, which always occurs when the mug's inner radius equals the inner depth (); the mug holds the most liquid possible with the given amount of material
Snapshot 4: the current and optimal design shown in a 2D contour plot where the liquid volume is the target function and the pink line is the constrained function, the mug material; the yellow point indicates the liquid volume for the current mug parameters and the red point is the optimal design
(For a closed container like an oil tank, the constrained function is slightly different. In that case the optimum occurs when .)

### 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 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.