2016 MCM Problem A A Hot Bath 热水浴
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.
一个人用热水从一个水龙头里灌满一个浴缸，然后躺在浴缸中，清洗和放松。不幸的是，这个浴缸不是一个具有二次加热系统和循环射流的温泉式浴缸，而是一个简单的水容器。过了一会儿，洗澡水就会明显变凉，所以这个人增加了一个可以恒定流出热水的水龙头来重新加热洗澡水。该浴缸的设计是：当浴缸达到最大容量时，多余的水可以通过溢漏排出去。
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. 在空间和时间上开发一个浴缸的水的温度模型，以确定在浴缸的人可以采取的最佳策略以来保持温度，并用最节省的水量来使整个浴缸的温度尽可能接近的初始温度。
使用你的模型来确定你的策略取决的浴缸的形状和体积，浴缸的形状/体积/温度，浴缸中的人的运动。如果这个人在最初填充浴缸时用了一个泡泡浴剂来协助洗澡，这会如何影响你的模型的结果？除了要求的一页摘要MCM提交，你的报告必须包括一页给用户的有关浴缸的非技术性的解释页，说明你的技术以及为什么洗澡水的温度很难达到均匀恒定。
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