2016年美赛A题美国大学生获奖论文

ForofficeuseonlyT1________________T2________________T3________________T4________________TeamControlNumber54164ProblemChosenAForofficeuseonlyF1________________F2________________F3________________F4________________2016MCM/ICMSummarySheet(Yourteam'ssummaryshouldbeincludedasthefirstpageofyourelectronicsubmission.)Typeasummaryofyourresultsonthispage.Donotincludethenameofyourschool,advisor,orteammembersonthispage.AbstractWepresentauser-friendlystrategyforkeepingthewaterinabathtubascloseaspossibletoacertainpreferredtemperature.Thisstrategyconsistsintwomainconstituents:adjustingthehotwaterflowfromthefaucettokeepthebathtubtemperaturesuitablyhigh,andusinghandmotionstodistributetheheatevenlythroughoutthebathwater.Asfordeterminingthenecessarywaterflow,wefirstcreateamodelthatdeterminesthetemperatureofthebathwateratanypointintime.Wepositthatthetemperatureofthewaterinthebathisdependentonthecoolingduetotheambienttemperatureoftheroom,whichobeystheproportionalitygivenbyNewton’sLawofCooling;andtheheatingduetotheaddedhotwaterfromthefaucet.Ourmodelcombinesthetemperaturechangesovertimefromthesetwoentitiesintooneconsummatedifferentialequation.WedeterminetheexperimentalconstantofproportionalityforNewton’sLawofCoolingbyproposingthatthisconstantitselfisdependentonacombinationofvolumeofthewaterandsurfaceareaexposedtotheair.Takingexperimentaldatafromvariedsources,includinganexperimentweconductedanddocumentedoverthetimeframeoftheMCMcompetition,wewereabletoextrapolatevaluesfortheconstantattributabletoanybathtubgivenitsvolumeandsurfaceareaexposedtoair.Wethenrunacomputersimulationtomodelthetemperaturechangesovertimeinastandardbathtubwithourdifferentialequation,determininghowmuchhotwaterneedstoflowintokeepthetotaldevianceovertimeofthetemperatureofthebathfromadesiredtemperatureataminimum.Wefoundtheoptimalflowrateforastandardbathtubatstandardconditionstobe1.81gallons/minute,orabout11%ofthemaximumflowrate.Ourmodelwasabletopredictoptimalflowratesformanydifferentcombinationsofvolume,surfacearea,tubmaterial,andadditives.Wefoundthatmoreeffectiveinsulatingmaterialsledtoloweroptimalflowrates(lesshotwaterneeded),asdidtheadditionofadditives.Volumeandsurfacearea,surprisingly,showednocleartrend.Asforkeepingthetemperatureroughlyeveninthebathtub,wesuggestthattheuserswirlhishandscounter-clockwisewithrespecttothefaucetasshowninFigure1.Thisisasimpleyeteffectivewaytodistributetheheatacceptablyevenlythroughoutthetubwater.Asalternativesforthosewhodonotwishtomaketheirbathintoanexercisesession,weinventedanoriginalproduct(HomeJacuzzi)thatislow-costandaidsinheatingthewaterevenly.TheMathBehindtheBath:ACoolInvestigationonCoolingTeam#54164February2016Team#54164Page1of421AbstractWepresentauser-friendlystrategyforkeepingthewaterinabathtubascloseaspossibletoacertainpreferredtemperature.Thisstrategyconsistsintwomainconstituents:adjustingthehotwaterowfromthefaucettokeepthebathtubtemperaturesuitablyhigh,andusinghandmotionstodistributetheheatevenlythroughoutthebathwater.Asfordeterminingthenecessarywaterow,werstcreateamodelthatdeterminesthetem-peratureofthebathwateratanypointintime.Wepositthatthetemperatureofthewaterinthebathisdependentonthecoolingduetotheambienttemperatureoftheroom,whichobeystheproportionalitygivenbyNewton'sLawofCooling;andtheheatingduetotheaddedhotwaterfromthefaucet.Ourmodelcombinesthetemperaturechangesovertimefromthesetwoentitiesintooneconsummatedierentialequation.WedeterminetheexperimentalconstantofproportionalityforNewton'sLawofCoolingbyproposingthatthisconstantitselfisdependentonacombinationofvolumeofthewaterandsurfaceareaexposedtotheair.Takingexperimentaldatafromvariedsources,includinganexperimentweconductedanddocumentedoverthetimeframeoftheMCMcompetition,wewereabletoextrapolatevaluesfortheconstantattributabletoanybathtubgivenitsvolumeandsurfaceareaexposedtoair.Wethenrunacomputersimulationtomodelthetemperaturechangesovertimeinastandardbathtubwithourdierentialequation,determininghowmuchhotwaterneedstoowintokeepthetotaldevianceovertimeofthetemperatureofthebathfromadesiredtemperatureataminimum.Wefoundtheoptimalowrateforastandardbathtubatstandardconditionstobe1.81gallons/minute,orabout11%ofthemaximumowrate.Ourmodelwasabletopredictoptimalowratesformanydierentcombinationsofvolume,surfacearea,tubmaterial,andadditives.Wefoundthatmoreeectiveinsulatingmaterialsledtoloweroptimalowrates(lesshotwaterneeded),asdidtheadditionofadditives.Volumeandsurfacearea,surprisingly,showednocleartrend.Asforkeepingthetemperatureroughlyeveninthebathtub,wesuggestthattheuserswirlhishandscounter-clockwisewithrespecttothefaucetasshowninFigure1.Thisisasimpleyeteectivewaytodistributetheheatacceptablyevenlythroughoutthetubwater.Asalternativesforthosewhodonotwishtomaketheirbathintoanexercisesession,weinventedanoriginalproduct(HomeJacuzzi)thatislow-costandaidsinheatingthewaterevenly.1.1Non-TechnicalExplanationWhyisitsodiculttokeepthetemperatureofthewatereventhroughoutwhentakingabath?Inastandardsingle-faucetbathtub,hotwatercomesoutofthefaucetononlyonesideofthebath,leavingtheotherside