英文-历届美国数学建模竞赛赛题(2001-2014)

2001MCMA:ChoosingaBicycleWheelCyclistshavedifferenttypesofwheelstheycanuseontheirbicycles.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:Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuationtrafficflow?Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Required:Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Required:Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.2002MCMA:WindandWatersprayAnornamentalfountaininalargeopenplazasurroundedbybuildingssquirtswaterhighintotheair.Ongustydays,thewindblowssprayfromthefountainontopassersby.Thewater-flowfromthefountainiscontrolledbyamechanismlinkedtoananemometer(whichmeasureswindspeedanddirection)locatedontopofanadjacentbuilding.Theobjectiveofthiscontrolistoprovidepassersbywithanacceptablebalancebetweenanattractivespectacleandasoaking:Theharderthewindblows,thelowerthewatervolumeandheighttowhichthewaterissquirted,hencethelesssprayfallsoutsidethepoolarea.Yourtaskistodeviseanalgorithmwhichusesdataprovidedbytheanemometertoadjustthewater-flowfromthefountainasthewindconditionschange.2002MCMB:AirlineOverbookingYou'reallpackedandreadytogoonatriptovisityourbestfriendinNewYorkCity.Afteryoucheckinattheticketcounter,theairlineclerkannouncesthatyourflighthasbeenoverbooked.Passengersneedtocheckinimmediatelytodetermineiftheystillhaveaseat.Historically,airlinesknowthatonlyacertainpercentageofpassengerswhohavemadereservationsonaparticularflightwillactuallytakethatflight.Consequently,mostairlinesoverbook-thatis,theytakemorereservationsthanthecapacityoftheaircraft.Occasionally,morepassengerswillwanttotakeaflightthanthecapacityoftheplaneleadingtooneormorepassengersbeingbumpedandthusunabletotaketheflightforwhichtheyhadreservations.Airlinesdealwithbumpedpassengersinvariousways.Somearegivennothing,somearebookedonlaterflightsonotherairlines,andsomearegivensomekindofcashorairlineticketincentive.Considertheoverbookingissueinlightofthecurrentsituation:LessflightsbyairlinesfrompointAtopointBHeightenedsecurityatandaroundairportsPassengers'fearLossofbillionsofdollarsinrevenuebyairlinestodateBuildamathematicalmodelthatexaminestheeffectsthatdifferentoverbookingschemeshaveontherevenuereceivedbyanairlinecompanyinordertofindanoptimaloverbookingstrategy,i.e.,thenumberofpeoplebywhichanairlineshouldoverbookaparticularflightsothatthecompany'srevenueismaximized.Insurethatyourmodelreflectstheissuesabove,andconsideralternativesforhandling“bumped”passengers.Additionally,writeashortmemorandumtotheairline'sCEOsummarizingyourfindingsandanalysis.2003MCMA:TheStuntPersonAnexcitingactionsceneinamovieisgoingtobefilmed,andyouarethestuntcoordinator!Astuntpersononamotorcyclewilljumpoveranelephantandlandinapileofcardboardboxestocushiontheirfall.Youneedtoprotectthestuntperson,andalsouserelativelyfewcardboardboxes(lowercost,notseenbycamera,etc.).Yourjobisto:determinewhatsizeboxestousedeterminehowmanyboxestousedeterminehowtheboxeswillbestackeddetermineifanymodificationstotheboxeswouldhelpgeneralizetodifferentcombinedweights(stuntperson&motorcycle)anddifferentjumpheightsNotethat,in“TomorrowNeverDies”,theJamesBondcharacteronamotorcyclejumpsoverahelicopter.2003MCMB:GammaKnifeTreatmentPlanningStereotacticradiosurgerydeliversasinglehighdoseofionizingradiationtoaradiographicallywell-defined,smallintracranial3Dbraintumorwithoutdeliveringanysignificantfractionoftheprescribeddosetothesurroundingbraintissue.Threemodalitiesarecommonlyusedinthisarea;theyarethegammaknifeunit,heavychargedparticlebeams,andexternalhigh-energyphotonbeamsfromlinearaccelerators.Thegammaknifeunitdeliversasinglehighdoseofionizingradiationemanatingfrom201cobalt-60unitsourcesthroughaheavyhelmet.All201beamssimultaneouslyintersectattheisocenter,resultinginaspherical(approximately)dosedistributionattheeffectivedoselevels.Irradiatingtheisocentertodeliverdoseistermeda“shot.”Shotscanberepresentedasdifferentspheres.Fourinterchangeableoutercollimatorhelmetswithbeamchanneldiametersof4,8,14,and18mmareavailableforirradiatingdifferentsizevolumes.Foratargetvolumelargerthanoneshot,multipleshotscanbeusedtocovertheentiretarget.Inpractice,mosttargetvolumesaretreatedwith1to15shots.Thetargetvolumeisabounded,three-dimensionaldigitalimagethatusuallyconsistsofmillionsofpoints.Thegoalofradiosurgeryistodepletetumorcellswhilepreservingnormalstructures.Sincetherearephysicallimitationsandbiologicaluncertaintiesinvolvedinthistherapyprocess,atreatmentplanneedstoaccountforallthoselimitationsanduncertainties.Ingeneral,anoptimaltreatmentplanisdesignedtomeetthefollowingrequirements.Minimizethedosegradientacrossthetargetvolume.Matchspecifiedisodosecontourstothetargetvolumes.Matchspecifieddose-volumeconstraintsofthetargetandcriticalorgan.Minimizetheintegraldosetotheentirevolumeofnormaltissuesororgans.Constraindosetospecifiednormaltissuepointsbelowtolerancedoses.Minimizethemaximumdosetocriticalvolumes.Ingammaunittreatmentplanning,wehavethefollowingconstraints:Prohibitshotsfromprotrudingoutsidethetarget.Prohibitshotsfromoverlapping(toavoidhotspots).Coverthetargetvolumewitheffectivedosageasmuchaspossible.Butatleast90%ofthetargetvolumemustbecoveredbyshots.Useasfewshotsaspossible.Yourtasksaretoformulatetheoptimaltreatmentplanningforagammaknifeunitasasphere-packingproblem,andproposeanalgorithmtofindasolution.Whiledesigningyouralgorithm,youmustkeepinmindthatyouralgorithmmustbereasonablyefficient.2004MCMA:AreFingerprintsUnique?Itisacommonplacebeliefthatthethumbprintofeveryhumanwhohaseverlivedisdifferent.Developandanalyzeamodelthatwillallowyoutoassesstheprobabilitythatthisistrue.Comparetheodds(thatyoufoundinthisproblem)ofmisidentificationbyfingerprintevidenceagainsttheoddsofmisidentificationbyDNAevidence.2004MCMB:AFasterQuickPassSystem“QuickPass”systemsareincreasinglyappearingtoreducepeople'stimewaitinginline,whetheritisattollbooths,amusementparks,orelsewhere.ConsiderthedesignofaQuickPasssystemforanamusementpark.TheamusementparkhasexperimentedbyofferingQuickPassesforseveralpopularridesasatest.Theideaisthatforcertainpopularridesyoucangotoakiosknearthatrideandinsertyourdailyparkentranceticket,andoutwillcomeaslipthatstatesthatyoucanreturntothatrideataspecifictimelater.Forexample,youinsertyourdailyparkentranceticketat1:15pm,andtheQuickPassstatesthatyoucancomebackbetween3:30and4:30pmwhenyoucanuseyoursliptoenterasecond,andpresumablymuchshorter,linethatwillgetyoutotheridefaster.TopreventpeoplefromobtainingQuickPassesforseveralridesatonce,theQuickPassmachinesallowyoutohaveonlyoneactiveQuickPassatatime.YouhavebeenhiredasoneofseveralcompetingconsultantstoimprovetheoperationofQuickPass.Customershavebeencomplainingaboutsomeanomaliesinthetestsystem.Forexample,customersobservedthatinoneinstanceQuickPasseswerebeingofferedforareturntimeaslongas4hourslater.Ashorttimelateronthesameride,theQuickPassesweregivenfortimesonlyanhourorsolater.Insomeinstances,thelinesforpeoplewithQuickpassesarenearlyaslongandslowastheregularlines.TheproblemthenistoproposeandtestschemesforissuingQuickPassesinordertoincreasepeople'senjoymentoftheamusementpark.Partoftheproblemistodeterminewhatcriteriatouseinevaluatingalternativeschemes.Includeinyourreportanon-technicalsummaryforamusementparkexecutiveswhomustchoosebetweenalternativesfromcompetingconsultants.2005MCMA:FloodPlanningLakeMurrayincentralSouthCarolinaisformedbyalargeearthendam,whichwascompletedin1930forpowerproduction.Modelthefloodingdownstreamintheeventthereisacatastrophicearthquakethatbreachesthedam.Twoparticularquestions:RawlsCreekisayear-roundstreamthatflowsintotheSaludaRiverashortdistancedownriverfromthedam.HowmuchfloodingwilloccurinRawlsCreekfromadamfailure,andhowfarbackwillitextend?CouldthefloodbesomassivedownstreamthatwaterwouldreachuptotheS.C.StateCapitolBuilding,whichisonahilloverlookingtheCongareeRiver?2005MCMB:TollboothsHeavily-traveledtollroadssuchastheGardenStateParkway,Interstate95,andsoforth,aremulti-lanedividedhighwaysthatareinterruptedatintervalsbytollplazas.Becausecollectingtollsisusuallyunpopular,itisdesirabletominimizemotoristannoyancebylimitingtheamountoftrafficdisruptioncausedbythetollplazas.Commonly,amuchlargernumberoftollboothsisprovidedthanthenumberoftravellanesenteringthetollplaza.Uponenteringthetollplaza,theflowofvehiclesfansouttothelargernumberoftollbooths,andwhenleavingthetollplaza,theflowofvehiclesisrequiredtosqueezebackdowntoanumberoftravellanesequaltothenumberoftravellanesbeforethetollplaza.Consequently,whentrafficisheavy,congestionincreasesupondeparturefromthetollplaza.Whentrafficisveryheavy,congestionalsobuildsattheentrytothetollplazabecauseofthetimerequiredforeachvehicletopaythetoll.Makeamodeltohelpyoudeterminetheoptimalnumberoftollboothstodeployinabarrier-tollplaza.Explicitlyconsiderthescenariowherethereisexactlyonetollboothperincomingtravellane.Underwhatconditionsisthismoreorlesseffectivethanthecurrentpractice?Notethatthedefinitionof“optimal”isuptoyoutodetermine.2006MCMA:PositioningandMovingSprinklerSystemsforIrrigationThereareawidevarietyoftechniquesavailableforirrigatingafield.Thetechnologiesrangefromadvanceddripsystemstoperiodicflooding.Oneofthesystemsthatisusedonsmallerranchesistheuseof“handmove”irrigationsystems.Lightweightaluminumpipeswithsprinklerheadsareputinplaceacrossfields,andtheyaremovedbyhandatperiodicintervalstoinsurethatthewholefieldreceivesanadequateamountofwater.Thistypeofirrigationsystemischeaperandeasiertomaintainthanothersystems.Itisalsoflexible,allowingforuseonawidevarietyoffieldsandcrops.Thedisadvantageisthatitrequiresagreatdealoftimeandefforttomoveandsetuptheequipmentatregularintervals.Giventhatthistypeofirrigationsystemistobeused,howcanitbeconfiguredtominimizetheamountoftimerequiredtoirrigateafieldthatis80metersby30meters?Forthistaskyouareaskedtofindanalgorithmtodeterminehowtoirrigatetherectangularfieldthatminimizestheamountoftimerequiredbyaranchertomaintaintheirrigationsystem.Onepipesetisusedinthefield.Youshoulddeterminethenumberofsprinklersandthespacingbetweensprinklers,andyoushouldfindascheduletomovethepipes,includingwheretomovethem.Apipesetconsistsofanumberofpipesthatcanbeconnectedtogetherinastraightline.Eachpipehasa10cminnerdiameterwithrotatingspraynozzlesthathavea0.6cminnerdiameter.Whenputtogethertheresultingpipeis20meterslong.Atthewatersource,thepressureis420Kilo-Pascal’sandhasaflowrateof150litersperminute.Nopartofthefieldshouldreceivemorethan0.75cmperhourofwater,andeachpartofthefieldshouldreceiveatleast2centimetersofwaterevery4days.Thetotalamountofwatershouldbeappliedasuniformlyaspossible.2006MCMB:WheelChairAccessatAirportsOneofthefrustrationswithairtravelistheneedtoflythroughmultipleairports,andeachstopgenerallyrequireseachtravelertochangetoadifferentairplane.Thiscanbeespeciallydifficultforpeoplewhoarenotabletoeasilywalktoadifferentflight'swaitingarea.Oneofthewaysthatanairlinecanmakethetransitioneasieristoprovideawheelchairandanescorttothosepeoplewhoaskforhelp.Itisgenerallyknownwellinadvancewhichpassengersrequirehelp,butitisnotuncommontoreceivenoticewhenapassengerfirstregistersattheairport.Inrareinstancesanairlinemaynotreceivenoticefromapassengeruntiljustpriortolanding.Airlinesareunderconstantpressuretokeeptheircostsdown.Wheelchairswearoutandareexpensiveandrequiremaintenance.Thereisalsoacostformakingtheescortsavailable.Moreover,wheelchairsandtheirescortsmustbeconstantlymovedaroundtheairportsothattheyareavailabletopeoplewhentheirflightlands.Insomelargeairportsthetimerequiredtomoveacrosstheairportisnontrivial.Thewheelchairsmustbestoredsomewhere,butspaceisexpensiveandseverelylimitedinanairportterminal.Also,wheelchairsleftinhightrafficareasrepresentaliabilityriskaspeopletrytomovearoundthem.Finally,oneofthebiggestcostsisthecostofholdingaplaneifsomeonemustwaitforanescortandbecomeslatefortheirflight.Thelattercostisespeciallytroublingbecauseitcanaffecttheairline'saverageflightdelaywhichcanleadtofewerticketsalesaspotentialcustomersmaychoosetoavoidanairline.EpsilonAirlineshasdecidedtoaskathirdpartytohelpthemobtainadetailedanalysisoftheissuesandcostsofkeepingandmaintainingwheelchairsandescortsavailableforpassengers.Theairlineneedstofindawaytoschedulethemovementofwheelchairsthroughouteachdayinacosteffectiveway.Theyalsoneedtofindanddefinethecostsforbudgetplanninginboththeshortandlongterm.EpsilonAirlineshasaskedyourconsultantgrouptoputtogetherabidtohelpthemsolvetheirproblem.Yourbidshouldincludeanoverviewandanalysisofthesituationtohelpthemdecideifyoufullyunderstandtheirproblem.Theyrequireadetaileddescriptionofanalgorithmthatyouwouldliketoimplementwhichcandeterminewheretheescortsandwheelchairsshouldbeandhowtheyshouldmovethroughouteachday.Thegoalistokeepthetotalcostsaslowaspossible.Yourbidisoneofmanythattheairlinewillconsider.Youmustmakeastrongcaseastowhyyoursolutionisthebestandshowthatitwillbeabletohandleawiderangeofairportsunderavarietyofcircumstances.Yourbidshouldalsoincludeexamplesofhowthealgorithmwouldworkforalarge(atleast4concourses),amedium(atleasttwoconcourses),andasmallairport(oneconcourse)underhighandlowtrafficloads.Youshoulddetermineallpotentialcostsandbalancetheirrespectiveweights.Finally,aspopulationsbegintoincludeahigherpercentageofolderpeoplewhohavemoretimetotravelbutmayrequiremoreaid,yourreportshouldincludeprojectionsofpotentialcostsandneedsinthefuturewithrecommendationstomeetfutureneeds.2007MCMA:GerrymanderingGerrymanderingTheUnitedStatesConstitutionprovidesthattheHouseofRepresentativesshallbecomposedofsomenumber(currently435)ofindividualswhoareelectedfromeachstateinproportiontothestate’spopulationrelativetothatofthecountryasawhole.Whilethisprovidesawayofdetermininghowmanyrepresentativeseachstatewillhave,itsaysnothingabouthowthedistrictrepresentedbyaparticularrepresentativeshallbedeterminedgeographically.Thisoversighthasledtoegregious(atleastsomepeoplethinkso,usuallynottheincumbent)districtshapesthatlook“unnatural”bysomestandards.Hencethefollowingquestion:Supposeyouweregiventheopportunitytodrawcongressionaldistrictsforastate.Howwouldyoudosoasapurely“baseline”exercisetocreatethe“simplest”shapesforallthedistrictsinastate?Therulesincludeonlythateachdistrictinthestatemustcontainthesamepopulation.Thedefinitionof“simple”isuptoyou;butyouneedtomakeaconvincingargumenttovotersinthestatethatyoursolutionisfair.Asanapplicationofyourmethod,drawgeographicallysimplecongressionaldistrictsforthestateofNewYork.2007MCMB:TheAirplaneSeatingProblemAirlinesarefreetoseatpassengerswaitingtoboardanaircraftinanyorderwhatsoever.Ithasbecomecustomarytoseatpassengerswithspecialneedsfirst,followedbyfirst-classpassengers(whositatthefrontoftheplane).Thencoachandbusiness-classpassengersareseatedbygroupsofrows,beginningwiththerowatthebackoftheplaneandproceedingforward.Apartfromconsiderationofthepassengers’waittime,fromtheairline’spointofview,timeismoney,andboardingtimeisbestminimized.Theplanemakesmoneyfortheairlineonlywhenitisinmotion,andlongboardingtimeslimitthenumberoftripsthataplanecanmakeinaday.Thedevelopmentoflargerplanes,suchastheAirbusA380(800passengers),accentuatetheproblemofminimizingboarding(anddeboarding)time.Deviseandcompareproceduresforboardinganddeboardingplaneswithvaryingnumbersofpassengers:small(85–210),midsize(210–330),andlarge(450–800).Prepareanexecutivesummary,nottoexceedtwosingle-spacedpages,inwhichyousetoutyourconclusionstoanaudienceofairlineexecutives,gateagents,andflightcrews.AnarticleappearedintheNYTimesNov14,2006addressingprocedurescurrentlybeingfollowedandtheimportancetotheairlineoffindingbettersolutions.Thearticlecanbeseenat:\o"/2006/11/14/business/14boarding.html"/2006/11/14/business/14boarding.html2008MCMA:TakeaBathConsidertheeffectsonlandfromthemeltingofthenorthpolaricecapduetothepredictedincreaseinglobaltemperatures.Specifically,modeltheeffectsonthecoastofFloridaeverytenyearsforthenext50yearsduetothemelting,withparticularattentiongiventolargemetropolitanareas.Proposeappropriateresponsestodealwiththis.Acarefuldiscussionofthedatausedisanimportantpartoftheanswer.2008MCMB:CreatingSudokuPuzzlesDevelopanalgorithmtoconstructSudokupuzzlesofvaryingdifficulty.Developmetricstodefineadifficultylevel.Thealgorithmandmetricsshouldbeextensibletoavaryingnumberofdifficultylevels.Youshouldillustratethealgorithmwithatleast4difficultylevels.Youralgorithmshouldguaranteeauniquesolution.Analyzethecomplexityofyouralgorithm.Yourobjectiveshouldbetominimizethecomplexityofthealgorithmandmeettheaboverequirements.2009MCMA:DesigningaTrafficCircleManycitiesandcommunitieshavetrafficcircles—fromlargeoneswithmanylanesinthecircle(suchasattheArcdeTriompheinParisandtheVictoryMonumentinBangkok)tosmalloneswithoneortwolanesinthecircle.Someofthesetrafficcirclespositionastopsignorayieldsignoneveryincomingroadthatgivesprioritytotrafficalreadyinthecircle;somepositionayieldsigninthecircleateachincomingroadtogiveprioritytoincomingtraffic;andsomepositionatrafficlightoneachincomingroad(withnorightturnallowedonaredlight).Otherdesignsmayalsobepossible.Thegoalofthisproblemistouseamodeltodeterminehowbesttocontroltrafficflowin,around,andoutofacircle.Stateclearlytheobjective(s)youuseinyourmodelformakingtheoptimalchoiceaswellasthefactorsthataffectthischoice.IncludeaTechnicalSummaryofnotmorethantwodouble-spacedpagesthatexplainstoaTrafficEngineerhowtouseyourmodeltohelpchoosetheappropriateflow-controlmethodforanyspecifictrafficcircle.Thatis,summarizetheconditionsunderwhicheachtypeoftraffic-controlmethodshouldbeused.Whentrafficlightsarerecommended,explainamethodfordetermininghowmanysecondseachlightshouldremaingreen(whichmayvaryaccordingtothetimeofdayandotherfactors).Illustratehowyourmodelworkswithspecificexamples.2009MCMB:EnergyandtheCellPhoneThisquestioninvolvesthe“energy”consequencesofthecellphonerevolution.Cellphoneusageismushrooming,andmanypeopleareusingcellphonesandgivinguptheirlandlinetelephones.Whatistheconsequenceofthisintermsofelectricityuse?Everycellphonecomeswithabatteryandarecharger.Requirement1ConsiderthecurrentUS,acountryofabout300millionpeople.EstimatefromavailabledatathenumberHofhouseholds,withmmemberseach,thatinthepastwereservicedbylandlines.Now,supposethatallthelandlinesarereplacedbycellphones;thatis,eachofthemmembersofthehouseholdhasacellphone