2016年(美赛)MCMABC题及核心意思提点

A题Apersonfillsabathtubwithhotwaterfromasinglefaucetandsettlesintothebathtubtocleanseandrelax.Unfortunately,thebathtubisnotaspa-styletubwithasecondaryheatingsystemandcirculatingjets,butratherasimplewatercontainmentvessel.Afterawhile,thebathgetsnoticeablycooler,sothepersonaddsaconstanttrickleofhotwaterfromthefaucettoreheatthebathingwater.Thebathtubisdesignedinsuchawaythatwhenthetubreachesitscapacity,excesswaterescapesthroughanoverflowdrain.Developamodelofthetemperatureofthebathtubwaterinspaceandtimetodeterminethebeststrategythepersoninthebathtubcanadopttokeepthetemperatureeventhroughoutthebathtubandascloseaspossibletotheinitialtemperaturewithoutwastingtoomuchwater.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。Useyourmodeltodeterminetheextenttowhichyourstrategydependsupontheshapeandvolumeofthetub,theshape/volume/temperatureofthepersoninthebathtub,andthemotionsmadebythepersoninthebathtub.Ifthepersonusedabubblebathadditive（气泡浴剂）whileinitiallyfillingthebathtubtoassistincleansing,howwouldthisaffectyourmodel’sresults?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？Inadditiontotherequiredone-pagesummaryforyourMCMsubmission,yourreportmustincludeaone-pagenon-technicalexplanationforusersofthebathtubthatdescribesyourstrategywhileexplainingwhyitissodifficulttogetanevenlymaintainedtemperaturethroughoutthebathwater.B题Theamountofsmalldebrisinorbitaroundearthhasbeenagrowingconcern.Itisestimatedthatmorethan500,000piecesofspacedebris,alsocalledorbitaldebris,arecurrentlybeingtrackedaspotentialhazardstospacecraft.TheissueitselfbecamemorewidelydiscussedinthenewsmediawhentheRussiansatelliteKosmos-2251andtheUSAsatelliteIridium-33collidedon10February,2009.Anumberofmethodstoremovethedebrishavebeenproposed.Thesemethodsincludesmall,space-basedwaterjetsandhighenergylasersusedtotargetspecificpiecesofdebrisandlargesatellitesdesignedtosweepupthedebris,amongothers.Thedebrisrangesinsizeandmassfrompaintflakestoabandonedsatellites.Thedebris’highvelocityorbitsmakecapturedifficult.Developatime-dependentmodeltodeterminethebestalternativeorcombinationofalternativesthataprivatefirmcouldadoptasacommercialopportunitytoaddressthespacedebrisproblem.Yourmodelshouldincludequantitativeand/orqualitativeestimatesofcosts,risks,benefits,aswellasotherimportantfactors.Yourmodelshouldbeabletoassessindependentalternativesaswellascombinationsofalternativesandbeabletoexploreavarietyofimportant“Whatif?”scenarios.建立一个由时间来决定的模型，以确定最好的备选或组合组合备选方案使得一家私人公司可以在处理太空废墟物问题上将其采纳为一种商业机会。你们的模型应当包括对成本、风险、效益以及其他重要因素的定量和（或）定性的估计。也要能够评估独立的备选方案以及组合备选方案，并考虑多种“如果。。。又会。。。”的情形。Usingyourmodel,determinewhetheraneconomicallyattractiveopportunityexistsornosuchopportunityispossible.Ifaviablecommercialopportunityexistsasanalternativesolution,provideacomparisonofthedifferentoptionsforremovingdebris,andincludeaspecificrecommendationastohowthedebrisshouldberemoved.Ifnosuchopportunityispossible,thenprovideinnovativealternativesforavoidingcollisions.使用你们的模型确定是否存在一个经济上吸引人的机会亦或者这样的机会是不可能的。若一切实可行的商业机会作为一种替代解决方法是存在的，那么就出去废墟物的不同方法提供一个比较。模型还要包括关于废墟物当如何出去的特别建议。Inadditiontotherequiredone-pagesummaryforyourMCMsubmission,yourreportmustincludeatwo-pageExecutiveSummarythatdescribestheoptionsconsideredandmajormodelingresults,andprovidesarecommendationforaparticularaction,combinationofactions,ornoaction,asappropriatefromyourwork.TheExecutiveSummaryshouldbewrittenforhighlevelpolicymakersandnewsmediaanalystswhodonothaveatechnicalbackground.C题TheGoodgrantFoundationisacharitableorganizationthatwantstohelpimproveeducationalperformanceofundergraduatesattendingcollegesanduniversitiesintheUnitedStates.Todothis,thefoundationintendstodonateatotalof$100,000,000(US100million)toanappropriategroupofschoolsperyear,forfiveyears,startingJuly2016.Indoingso,theydonotwanttoduplicatetheinvestmentsandfocusofotherlargegrantorganizationssuchastheGatesFoundationandLuminaFoundation.YourteamhasbeenaskedbytheGoodgrantFoundationtodevelopamodeltodetermineanoptimalinvestmentstrategythatidentifiestheschools,theinvestmentamountperschool,thereturnonthatinvestment,andthetimedurationthattheorganization’smoneyshouldbeprovidedtohavethehighestlikelihoodofproducingastrongpositiveeffectonstudentperformance.Thisstrategyshouldcontaina1toNoptimizedandprioritizedcandidatelistofschoolsyouarerecommendingforinvestmentbasedoneachcandidateschool’sdemonstratedpotentialforeffectiveuseofprivatefunding,andanestimatedreturnoninvestment(ROI)definedinamannerappropriateforacharitableorganizationsuchastheGoodgrantFoundation.你们的团队受GF的委托，去建立一个模型来确定最优的投资策略包括确定学校、每个学校的投入、投资的回报以及该机构所提供的钱对学生学业起到一个强大的正效用所需要的时间跨度。你的策略要包括一个一对多的最优及你推荐学校的优先次序的投资，该投资需基于每个备选学校有效率的使用自有资金所表现得潜力，以及以一种适合诸如GF一类的慈善机构所定义的关于投资的预期回报。Toassistyoureffort,theattacheddatafile(ProblemCDATA.zip)containsinformationextractedfromtheU.S.NationalCenteronEducationStatistics(),whichmaintainsanextensivedatabaseofsurveyinformationonnearlyallpost-secondarycollegesanduniversitiesintheUnitedStates,andtheCollegeScorecarddataset(