2016年(美赛)MCMABC题和核心意思提点.doc

A题 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. 建立时间和空间上的关于浴缸水温度的模型，确定浴缸中的人可以采纳的最佳策略使得即便是人在浴缸中也能保持水温，且在不浪费水的同时还尽可能的接近原来的水温。 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（气泡浴剂）while initially filling the bathtub to assist in cleansing, how would this affect your model’s results? 用你的模型确定浴缸形状和体积、浴缸中人的形状/体积/温度以及浴缸中人的动作对你的策略的影响程度。如果人在最初水填入浴缸时使用了气泡浴剂，则对模型的结果又有何影响？ In addition to the required onesummary for your MCM submission, your report must include a onenon-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. B题 The amount of small debris in orbit around earth has been a growing concern. It is estimated that more than 500,000 pieces of space debris, also called orbital debris, are currently being tracked as potential hazards to space craft. The issue itself became more widely discussed in the news media when the Russian satellite Kosmos-2251 and the USA satellite Iridium-33 col