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Problem Title: A Hot Bath

  Year: 2016      
  Student Level: Undergraduate      
  Source: MCM      
  Commentary: Yes (1)      
  Student Papers: Yes (6)      

A person fills a bathtub with hot water from a single faucet and settles into the bathtub to cleanse and relax. Unfortunately, the bathtub is not a spa-style tub with a secondary heating system and circulating jets, but rather a simple water containment vessel. After a while, the bath gets noticeably cooler, so the person adds a constant trickle of hot water from the faucet to reheat the bathing water. The bathtub is designed in such a way that when the tub reaches its capacity, excess water escapes through an overflow drain.

Develop a model of the temperature of the bathtub water in space and time to determine the best strategy the person in the bathtub can adopt to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water.

Use your model to determine the extent to which your strategy depends upon the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub. If the person used a bubble bath additive while initially filling the bathtub to assist in cleansing, how would this affect your model's results?

In addition to the required one-page summary for your MCM submission, your report must include a one-page non-technical explanation for users of the bathtub that describes your strategy while explaining why it is so difficult to get an evenly maintained temperature throughout the bath water.


Judges' Commentary: Hot Bath Problem

Kathleen M. Shannon
Dept. of Mathematics and Computer Science,
Salisbury University, Salisbury, MD

  Student Papers      

Team 42221: Have a hot bath

Northwestern Polytechnical University, China


Team 44398: Enjoy a Cozy and Green Bath

Chongqing University, China


Team 44845: Beneath the Surface: A Thermal-Fluid Analysis of a Hot Bath

University of Colorado Boulder, CO, USA


Team 46830: Keep It Warm

Southwest Jiaotong University, China


Team 48649: A Hot Bath

Shanghai Jiao Tong University, China


Team 54164: The MathBehindtheBath: A "Cool" Investigationon Cooling

Adlai E. Stevenson High School, IL, USA