2016美賽賽題官方解讀已打

1、Judges Commentary277Judges Commentary: Hot BathProblemKathleen M. ShanDept. of Mathematics and Computer Science, Salisbury University, Salisbury, MDroductionThe teams who chose the Hot Bath Problemhe 2016 MCM were asked to find an optimal strategy for taking a hot bath in a tradit

2、ionaltub,t is,ith no circulating jets. This was a deceptively difficultproblem to address. Of course, there are many established ms for heattransfer and fluid flow, and we saw a number of p equation and Navier-Stokes equations; but also, asrs using the heat ould expect in anundergraduate competition

3、, we saperst simply used Newtons lawof cooling. Many undergraduates do not see partial differential equationshe course of their study, and most of the judges took thiso accountwhen evaluating the prs.The judging pros itself should be oferest to teams and advisors; butsince is been described inin pre

4、vious judges commentaries (seeBlack 2009; 2011; 2013), I do not provide an overview here. In particular,the pros sheds light on the importance of various components in solutionprs.his commentary, I will focus primarily on specifics as theypertao this problemeneral advice to teams.Graphics, Simulatio

5、ns, and MWe saw a number of simulations and graphical represensions of heatflow. The bettrs explained well what the graphics were showing andhow they related to the mand the teams recommendations. Theseprs also gave a good description of the algorithm used to create thegraphics. All too many prs pre

6、sent graphics, and sometimes includecode in an appendix, without giving the judgefficient information toThe UMAP Journal 37 (3) (2016) 277281. c Copyright 2016 by COMAP, Inc.s.Permisto make digital or hard copies of part or all of this work foral or classroom useis granted without fee providedt copi

7、es are not made or distributed for profit or commerladvantage andt copies bear this notice. Abstracting with credit is permitted, bopyrightsfor components of this work owned by othersn COMAP must be honored. To copy otherwise,to republish, tot on servers, or to redistribute to lists requires prior p

8、ermisfrom COMAP.278The UMAP Journal37.3 (2016)evaluate them. Judgese neither the time nor theerest in readingcode in order to discern the algorithm. Descriptive captions, along withclear exition of how the graphics were created, are a must.The higher-red prs included mathematicst was appropriateand

9、justified. If there was a simulation, it was clear how the simulation wascreated and there was a clear justification of the mused. Bettrslimited the number of mst they used in order to flesh out fully thes on the ones they did use.CommunicationThe value of clear communicationhis competition cannot b

10、e overlystressed. Judgese limited time to read the pr, and the most brilliantming will fall by the wayside if thes are not communicated clearlyand efficiently.In particular, the project summary should outline the mused andthe results. For this problem, it shoulde indicated the best strategy forkeng

11、the water hot, what mwas used, and how the mled tothis strategy.When data, ms, equations,raphics are taken from other sour,appropriateation is required! But teams also need to discuss whythis m, equation, data, etc., is relevant and exactly how they adapted itto fit this problem.TheThe-Technical Exp

12、lanation for Users-technical explanationst are frequently asked for-ing competitions are important and often not given enough attention byparticipants. The best mathematics is not going to be helpfulhe “realworld” unless you can convince a-technical audiencet the resultsare valid. And the results wi

13、ll be totally useless if they cannot be translatedo indications of what the “cnt” should actually do with them.his case, what should the bather do to optimize enjoyment of thebath? There was some room for originality and creativity here. Many of thebest prs specifically recognizedt you somehow need

14、to stir the water,andt adding hot water to the top of the tub while the overflow valve isalso at the top is going to present problems. Some prs recommendeddraining part of the cooler water before adding new hot water, and othersdevised mechanisms for moving the hot water to the bottom of the tub.Jud

15、ges Commentary279BathtubsTeams regardedt looked at different sizes and shs of tubs were more highlyn thoset didnt. It was also expectedt the tub would bemedhree dimens, not just as a series of two-dimenal sli.The best prs had three-dimenal ms and discussed the effect ofthe shand size of the tub had

16、on their ms.Ming vs. Applying Ms CorrectlyOne of the moreeresting tenshis years judging was betndeveloand/or adapting ms as oped to finding and appropri-ay using existing ms.As mentioned previously, there are many accepted ms and differ-ential equations available to describe heat transfer and fluid

17、flow. Amongp mrst used these, the better ones were explicit in sing where thes came from, how and why they were appd, and what results wereobtained from their use.One of the impressive thingst we saw ie of the Outstandingprs was a creative solution to the need for appropriate parameters for these ms

18、in particular the proportionality constant k in Newtons lawof cooling. The team found souronlinet gave the values for k forwater in beakers; but they needed to mwater in a variety of shsof bathtubs. They performed experiments to measure the proportional-ity constant for a particular bathtub, then us

19、ed a variety of curve-fitting techniques to estimate the parameter value for tubs of other sizes. (Experi- ments in general, when appropriate, are always a good idea and provide a welcome respite for the judges from textbook approaches.)The curve-fittingt the team used was informed byyzing the s wou

20、lddimens involved, making amptions about which dimenimpact the value of k, and making suret the units would cancel out inthe curvest they appd. Unfortunay, thiwas not perfect. Infact, they had Newtons law of cooling wrong! Because of this, their initialms prediction was contrary to common sensethe t

21、emperature of thebath-water went to freezing instead of to room temperature. Of course,the team shoulde realizedt this was because they had the rate ofchange of temperature proportional to the temperature of the water rather n proportional to the difference betn the temperature of the waterand the t

22、emperature of the room. They did realizet the predictionswere ond adjusted the output of the mappropriay, arriving at thesolution to the differential equation the first place.t they shoulde been solving ine of the prst we see are everfect. Teams, after all,eonly akend to come up wim, solve it, enhan

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25、；698 元（三選二課程）798 元（全選課程）【限時特惠 568-668-768】課程+比賽四天指導(dǎo)+模擬賽+輔助報名（100）先在傳課上選擇普 通班課程進(jìn)行報名，報名后再支付 400 元到支付寶：， 支付后，請聯(lián)系下面進(jìn)入內(nèi)部群（比賽指導(dǎo)將通過進(jìn)行指導(dǎo)，大家務(wù)必迅速加進(jìn)去）VIP 班（算法班課程）課程+比賽四天指導(dǎo)+模擬賽+輔助報名（100）VIP 班（歷年賽題分析班）課程+比賽四天指導(dǎo)+模擬賽+輔助報名（100）280The UMAP Journal37.3 (2016)a clear and coherent p written a textbook willr presenting

26、their results. Anyho has everl you about still finding mistakes (with hopetthey are not substantive!) after years of use.So one of the most difficult, and sometimes contentious, decistthe judgese to make each year is what constitutes a “fatal” flaw in apr. What error is so gravet, in and of itself,

27、it would eliminate anotherwise Outstanding pr from receiving the Outstanding designation?This year it was the error noted above in Newtons law of coolingt ledto the most heated debate. Some judges feltt the team had redeemedthemselve the other mfficiently, in how they handled the break from reality

28、andingt they did, to allow us to overlook this flaw; whileothers disagreed, bevingt the team shoulde recognizedt thedifferential equations mused did not make good sense. This was arare case where, after lengthy discus, a consensus was not reached,and the pr was awarded the Outstanding designation by

29、 majority vote.However, judges on both sides of the controversy recognized the validity of the argument for the other side.The take-away from this for teams in future competitions is to double-or triple-check your established ms.In particular, when an estab-lished mseems to predict somethingt you kn

30、ow is wrong, questionwhether youe made a mistakehe m. This small error almostcost thithe Outstanding designation.The Hot Bath Problem allowed for a variety of approaches and a fairamount of creativity, even though most prs used established msfor heat and fluid flow. Teams who thought about the actua

31、l situation oftaking a bath rose above those who merely took the established ms andpresented results. As always, communication was key in determining thetop prs.Overall, there has been an improvementhe quality of the prst we see in final judging. Some of this is certainly due to the increasedpopular

32、ity of the contest and the smallercentage of prst makeit this far. However, it also seemst teams are making better use of theadvice and materials available to them.In particular, we are seeing more teams perform meaningful sensitivityysis. We are also seeing better use of amptionshe mingpros.More are making amptionstsimplify their ms andlater testing the sensitivity of their results to those amptions.All of the teams who sucsfully participatedhis years competitionshould be proud of their accomplishment. For those who wish to im