英文历届美国数学建模竞赛赛题.docx
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英文历届美国数学建模竞赛赛题.docx
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英文历届美国数学建模竞赛赛题
2001MCMA:
ChoosingaBicycleWheel
Cyclistshavedifferenttypesofwheelstheycanuseontheirbicycles.Thetwobasictypesofwheelsarethoseconstructedusingwirespokesandthoseconstructedofasoliddisk(seeFigure1)Thespokedwheelsarelighter,butthesolidwheelsaremoreaerodynamic.Asolidwheelisneverusedonthefrontforaroadracebutcanbeusedontherearofthebike.
Professionalcyclistslookataracecourseandmakeaneducatedguessastowhatkindofwheelsshouldbeused.Thedecisionisbasedonthenumberandsteepnessofthehills,theweather,windspeed,thecompetition,andotherconsiderations.Thedirectorsportifofyourfavoriteteamwouldliketohaveabettersysteminplaceandhasaskedyourteamforinformationtohelpdeterminewhatkindofwheelshouldbeusedforagivencourse.
Figure1:
Asolidwheelisshownontheleftandaspokedwheelisshownontheright.
Thedirectorsportifneedsspecificinformationtohelpmakeadecisionandhasaskedyourteamtoaccomplishthetaskslistedbelow.Foreachofthetasksassumethatthesamespokedwheelwillalwaysbeusedonthefrontbutthereisachoiceofwheelsfortherear.
∙Task1.Provideatablegivingthewindspeedatwhichthepowerrequiredforasolidrearwheelislessthanforaspokedrearwheel.Thetableshouldincludethewindspeedsfordifferentroadgradesstartingfromzeropercenttotenpercentinonepercentincrements.(Roadgradeisdefinedtobetheratioofthetotalriseofahilldividedbythelengthoftheroad.Ifthehillisviewedasatriangle,thegradeisthesineoftheangleatthebottomofthehill.)Ariderstartsatthebottomofthehillataspeedof45kph,andthedecelerationoftheriderisproportionaltotheroadgrade.Ariderwillloseabout8kphforafivepercentgradeover100meters.
∙Task2.Provideanexampleofhowthetablecouldbeusedforaspecifictimetrialcourse.
∙Task3.Determineifthetableisanadequatemeansfordecidingonthewheelconfigurationandofferothersuggestionsastohowtomakethisdecision.
2001MCMB:
EscapingaHurricane'sWrath(AnIllWind...)
EvacuatingthecoastofSouthCarolinaaheadofthepredictedlandfallofHurricaneFloydin1999ledtoamonumentaltrafficjam.TrafficslowedtoastandstillonInterstateI-26,whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafehavenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrivetookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.Fortunately,Floydturnednorthandsparedthestatethistime,butthepublicoutcryisforcingstateofficialstofindwaystoavoidarepeatofthistrafficnightmare.
TheprincipalproposalputforthtodealwiththisproblemisthereversaloftrafficonI-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfromCharlestontoColumbia.Planstocarrythisouthavebeenprepared(andpostedontheWeb)bytheSouthCarolinaEmergencyPreparednessDivision.TrafficreversalonprincipalroadsleadinginlandfromMyrtleBeachandHiltonHeadisalsoplanned.
AsimplifiedmapofSouthCarolinaisshown.Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongtherestofthecoastalstrip.(Moreaccuratedata,ifsought,arewidelyavailable.)
Theinterstateshavetwolanesoftrafficineachdirectionexceptinthemetropolitanareaswheretheyhavethree.Columbia,anothermetroareaofaround500,000people,doesnothavesufficienthotelspacetoaccommodatetheevacuees(includingsomecomingfromfarthernorthbyotherroutes),sosometrafficcontinuesoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20easttoAtlanta.In1999,trafficleavingColumbiagoingnorthwestwasmovingonlyveryslowly.Constructamodelfortheproblemtoinvestigatewhatstrategiesmayreducethecongestionobservedin1999.Herearethequestionsthatneedtobeaddressed:
1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?
2.In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?
3.SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?
4.WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?
5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?
6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuationtrafficflow?
Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Required:
Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.
Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.
Required:
Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.
2002MCMA:
WindandWaterspray
Anornamentalfountaininalargeopenplazasurroundedbybuildingssquirtswaterhighintotheair.Ongustydays,thewindblowssprayfromthefountainontopassersby.Thewater-flowfromthefountainiscontrolledbyamechanismlinkedtoananemometer(whichmeasureswindspeedanddirection)locatedontopofanadjacentbuilding.Theobjectiveofthiscontrolistoprovidepassersbywithanacceptablebalancebetweenanattractivespectacleandasoaking:
Theharderthewindblows,thelowerthewatervolumeandheighttowhichthewaterissquirted,hencethelesssprayfallsoutsidethepoolarea.
Yourtaskistodeviseanalgorithmwhichusesdataprovidedbytheanemometertoadjustthewater-flowfromthefountainasthewindconditionschange.
2002MCMB:
AirlineOverbooking
You'reallpackedandreadytogoonatriptovisityourbestfriendinNewYorkCity.Afteryoucheckinattheticketcounter,theairlineclerkannouncesthatyourflighthasbeenoverbooked.Passengersneedtocheckinimmediatelytodetermineiftheystillhaveaseat.
Historically,airlinesknowthatonlyacertainpercentageofpassengerswhohavemadereservationsonaparticularflightwillactuallytakethatflight.Consequently,mostairlinesoverbook-thatis,theytakemorereservationsthanthecapacityoftheaircraft.Occasionally,morepassengerswillwanttotakeaflightthanthecapacityoftheplaneleadingtooneormorepassengersbeingbumpedandthusunabletotaketheflightforwhichtheyhadreservations.
Airlinesdealwithbumpedpassengersinvariousways.Somearegivennothing,somearebookedonlaterflightsonotherairlines,andsomearegivensomekindofcashorairlineticketincentive.
Considertheoverbookingissueinlightofthecurrentsituation:
LessflightsbyairlinesfrompointAtopointBHeightenedsecurityatandaroundairportsPassengers'fearLossofbillionsofdollarsinrevenuebyairlinestodate
Buildamathematicalmodelthatexaminestheeffectsthatdifferentoverbookingschemeshaveontherevenuereceivedbyanairlinecompanyinordertofindanoptimaloverbookingstrategy,i.e.,thenumberofpeoplebywhichanairlineshouldoverbookaparticularflightsothatthecompany'srevenueismaximized.Insurethatyourmodelreflectstheissuesabove,andconsideralternativesforhandling“bumped”passengers.Additionally,writeashortmemorandumtotheairline'sCEOsummarizingyourfindingsandanalysis.
2003MCMA:
TheStuntPerson
Anexcitingactionsceneinamovieisgoingtobefilmed,andyouarethestuntcoordinator!
Astuntpersononamotorcyclewilljumpoveranelephantandlandinapileofcardboardboxestocushiontheirfall.Youneedtoprotectthestuntperson,andalsouserelativelyfewcardboardboxes(lowercost,notseenbycamera,etc.).
Yourjobisto:
∙determinewhatsizeboxestouse
∙determinehowmanyboxestouse
∙determinehowtheboxeswillbestacked
∙determineifanymodificationstotheboxeswouldhelp
∙generalizetodifferentcombinedweights(stuntperson&motorcycle)anddifferentjumpheights
Notethat,in“TomorrowNeverDies”,theJamesBondcharacteronamotorcyclejumpsoverahelicopter.
2003MCMB:
GammaKnifeTreatmentPlanning
Stereotacticradiosurgerydeliversasinglehighdoseofionizingradiationtoaradiographicallywell-defined,smallintracranial3Dbraintumorwithoutdeliveringanysignificantfractionoftheprescribeddosetothesurroundingbraintissue.Threemodalitiesarecommonlyusedinthisarea;theyarethegammaknifeunit,heavychargedparticlebeams,andexternalhigh-energyphotonbeamsfromlinearaccelerators.
Thegammaknifeunitdeliversasinglehighdoseofionizingradiationemanatingfrom201cobalt-60unitsourcesthroughaheavyhelmet.All201beamssimultaneouslyintersectattheisocenter,resultingin
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- 英文 历届 美国 数学 建模 竞赛