美國(MCM)(ICM)試題(1996年-2013年)(英文版)

美國大學(xué)生數(shù)學(xué)建模競賽試題

1996AmericanMCMProblemsProblemA

Theworld'soceanscontainanambientnoisefield.Seismicdisturbances,surface

shipping,andmarinemammalsaresourcesthat,indifferentfrequencyranges,contribute

tothisfield.Wewishtoconsiderhowthisambientnoisemightbeusedtodetectlarge

movingobjects,e.g.,submarineslocatedbelowtheoceansurface.Assumingthata

submarinemakesnointrinsicnoise,developamethodfordetectingthepresenceofa

movingsubmarine,itssize,anditsdirectionoftravel,usingonlyinformationobtained

bymeasuringchangestotheambientnoisefield.Beginwithnoiseatonefixedfrequency

andamplitude.

ProblemB

WhendeterminingthewinnerofacompetitionliketheMathematicalContestinModeling,

therearegenerallyalargenumberofpaperstojudge.Let/ssaythereareP=100papers.

AgroupofJjudgesiscollectedtoaccomplishthejudging.Fundingforthecontest

constainsboththenumberofjudgesthatcanbeobtainedandamountoftimethatthey

canjudge.ForeampleifP=100,thenj=8istypical.Ideally,eachjudgewouldreadpaper

andrank-orderthem,buttherearetoomanypapersforthis.Instead,therewillbea

numberofscreeningroundsinwhicheachjudgewillreadsomenumberofpapersandgive

themscores.Thensomeselectionschemeisusedtoreducethenumberofpapersunder

consideration:Ifthepapersarerank-ordered,thenthebottom30%thateachjudge

rank-orderscouldberejected.Alternatively,ifthejudgesdonotrank-order,but

insteadgivethemnumericalscore(say,from1to100),thenallpapersbelowsomecut-off

levelcouldberejected.

Thenewpoolofpapersisthenpassedbacktothejudges,andtheprocessisrepeated.

Aconcernisthenthetotalnumberofpapersthatjudgereadsmustbesubstantiallyless

thanP.TheprocessisstoppedwhenthereareonlyWpapersleft.Therearethewinners.

TypicallyforP=100,W=3.

Yourtaskistodetermineaselectionscheme,usingacombinationofrank-ordering,

numericalscoring,andothermethods,bywhichthefinalWpaperswi11includeonly

papersfromamongthe〃bcst〃2Wpapers.(By〃best〃，wcassumethatthereisanabsolute

rank-orderingtowhichalljudgeswouldagree.)Forexample,thetopthreepapers.Among

allsuchmethods,theonethatrequiredeachjudgetoreadtheleastnumberofpapers

isdesired.

Notethepossibilityofsystematicbiasinanumericalscoringscheme.Forexample,

foraspecificcollectionofpapers,onejudgecouldaverage70points,whileanother

couldaverage80points.Howwouldyouscaleyourschemetoaccommodateforchangesin

thecontestparameters(P,J,andW)?

1997AmericanMCMProblems

ProblemATheVelociraptorProblem

Thevelociraptor,Velociraptormongoliensis,wasapredatorydinosaurthatlivedduring

thelateCretaceousperiod,approximately75millionyearsago.Paleontologiststhink

thatitwasaverytenacioushunter,andmayhavehuntedinpairsorlarger

packs.Unfortunately,thereisnowaytoobserveitshuntingbehaviorinthewildas

canbedonewithmodernmammalianpredators.Agroupofpaleontologistshasapproached

yourteamandaskedforhelpinmodelingthehuntingbehaviorofthevelociraptor.They

hopetocompareyourresultswithfielddatareportedbybiologistsstudyingthe

behaviorsof1ions,tigers,andsimilarpredatoryanimals.

Theaverageadultvelociraptorwas3meterslongwithahipheightof0.5metersand

anapproximatemassof45kg.Itisestimatedthattheanimalcouldrunextremelyfast

atspeedof60km/hrforabout15seconds.Aftertheinitialburstofspeed,theanimal

neededtostopandrecoverfromabuildupoflacticacidinitsmuscles.

SupposethatvelociraptorpreyedonThescelosaurusneglectus,aherbivorousbiped

approximatelythesamesizeastheVelociraptor.Abiomachanicalanalysisofa

fossilizedThescelosaurusindicatesthatitcouldrunataspeedofabout50km/hr.

forlongperiodoftime.

Parti

Assumingthevelociraptorisasolitaryhunter,designamathematicalmodelthat

describeahuntingstrategyforasinglevelociraptorstalkingandchasingasingle

Thescelosaurusaswellastheevasivestrategyoftheprey.Assumethatthe

Thescelosauruscanalwaysdetectthevelociraptorwhenitcomeswithin15meters.but

maydetectthepredatoratevengreaterranges(upto50metersdependinguponthe

habitatandweatherconditions.Additionally,duetoitsphysicalstructureand

strength,thevelociraptorhasalimitedturningradiuswhenrunningatfullspeed.This

radiusisestimatedtobethreetimestheanimal'shipheight.Ontheotherhand,the

Thescelosaurusisextremelyagileandhasaturningradiusof0.5meters.

Part2

Assumingmorerealisticallythatthevelociraptorhuntedinpairs,designanewmodel

thatdescribesahuntingstrategyfortwovelociraptorstalkingandchasingasingle

Thescelosaurusaswellastheevasivestrategyoftheprey.Usetheotherassumptions

andlimitationsgiveninPart1.

ProblemBMixWellForFruitfulDiscussions

Smallgroupmeetingforthediscussionsofimportantissues,particularlong-range

planning,aregainingpopularity.Itisbelievedthatlargegroupsdiscourage

productivediscussionandthatadominantpersonalitywillusuallycontrolanddirect

thediscussion.Thus,incorporateboardmeetingstheboardwi11meetinsmal1groups

todiscussissuesbeforemeetingasawhole,thesesmallergroupsstilltuntherisk

ofcontrolbyadominantpersonality.Inanattempttoreducethisdangeritiscommon

toscheduleseveralsessionswithadifferentmixofpeopleineachgroup.

AmeetingofAnTostalCorporationwillbeattendedby29BoardMembersofwhichnine

arein-housemembers(i.e.,corporateemployees).Themeetingistobeanall-dayaffair

withthreesessionsscheduledforthemorningandfourfortheafternoon.Eachsession

willtake45minutes,beginningonthehourfrom9:00A.M.to4:00P.M.,withlunch

scheduledatnoon.Eachmorningsessionwillconsistofsixdiscussiongroupswitheach

discussiongroupledbyoneofthecorporation'ssixseniorofficers.Noneofthese

officersareboardmembers.Thuseachseniorofficerswillnotbeinvolvedinthe

afternoonsessionsandeachofthesesessionswillconsistofonlyfourdifferent

discussiongroups.

Thepresidentofthecorporationwantsalistofboard-memberassignmenttodiscussion

groupsforeachofthesevensessions.Theassignmentsshouldachieveasmuchofa

mixofthemembersasmuchaspossible.Theidealassignmentwouldhaveeachboardmember

inadiscussiongroupthesamenumberoftimeswhileminimizingcommonmembershipof

groupsforthedifferentsessions.

Theassignmentshouldalsosatisfythefollowingcriteria:

1.Forthemorningsessions,noboardmembershouldbeinthesameseniorofficer，s

discussiongrouptwice.

2.Nodiscussiongroupshouldcontainadisproportionatenumberofin-housemembers.

Givealistofassignmentsformembers1-9and10-29andofficers1-6.Indicatehowwell

thecriteriainthepreviousparagraphsaremet.Sinceitispossiblethatsomeboard

memberswillcancelatthelastminuteorthatsomenotscheduledwillshowup,an

algorithmthatthesecretarycouldusetoadjusttheassignmentswithanhour'snotice

wouldbeappreciated.Itwouldbeidealifthealgorithmcouldalsobeusedtomake

assignmentsforfuturemeetingsinvolvingdifferentlevelsofparticipationforeach

typeofattendee.

1998AmericanMCMProblems

ProblemAMRIScanners

Introduction

IndustrialmedicaldiagnosticmachinesknownasMagneticResonanceImager(MRI)scan

athree-dimensionalobjectsuchasabrain,anddelivertheirresultsintheformof

athree-dimensionalarrayofpixel.Eachpixelconsistsofonenumberindicatingacolor

orashadeofgraythatencodesameasureofwaterconcentrationinasmallregionof

thescannedobjectatthelocationofthepixel.Forinstance,0canpicturehighwater

concentrationinblack(ventricles,bloodvessels),128canpictureamediumwater

concentrationingray(brainnucleiandgraymatter),and255canpicturealowwater

densityinwhite(liquid-richwhitematterconsistingofmyelinatedaxons).SuchMRI

scannersalsoincludefacilitiestopictureonascreenanyhorizontalorverticalslice

throughthethree-dimensionalarray(slicesareparalleltoanyofthethreeCartesian

coordinateaxes).Algorithmsforpicturingslicesthroughobiiqueplanes,however,are

proprietary.Currentalgorithmsarelimitedintermsoftheanglesandparameteroptions

available;areimplementedonlyonheavilyuseddedicatedworkstations;lackinput

capabilitiesformarkingpointsinthepicturebeforeslicing;andtendtoblurand

“featherout〃sharpboundariesbetweentheoriginalpixels.

Amorefaithful,flexiblealgorithmimplementedonapersonalcomputerwouldbeuseful.

(1)forplanningminimallyinvasivetreatments,

(2)forcalibratingtheMRImachines,

(3)forinvestigatingstructuresorientedobliquelyinspace,suchaspost-mortemtissue

sectionsinaanimalresearch,

(4)forenablingcross-sectionsatanyanglethroughabrainatlasconsisting(4)for

enablingcross-sectionsatanyanglethroughabrainatlasconsisting

ofblack-and-whitelinedrawing

Todesignsuchanalgorithm,onecanaccessthevalueandlocationsofthepixels,but

nottheinitialdatagatheredbythescanners.

Problem

Designandtestanalgorithmthatproducessectionsofthree-dimensionalarraysby

planesinanyorientationinspace,preservingtheoriginalgray-scalevalueasclosely

aspossible.

DataSets

Thetypicaldatasetconsistsofathree-dimensionalarrayAofnumbersA(i,j,k)which

indicatesthedensityA(i,j,k)oftheobjectatthelocation(x,y,z)i,j,k.Typically

A(i,j,k)canrangefrom0to255.Inmostapplicationsthedatasetisquitelarge.

Teamsshoulddesigndatasetstotestanddemonstratetheiralgorithms.Thedatasets

shouldreflectconditionslikelyTeamsshoulddesigndatasetstotestanddemonstrate

theiralgorithms.Thedatasetsshouldreflectconditionslikelytobeofdiagnostic

interest.Teamsshouldalsocharacterizedatasetsthelimittheeffectivenessoftheir

algorithms.

Summary

Thealgorithmmustproduceapictureofthesliceofthethree-dimensionalarrayby

aplaneinspace.Theplanecanhaveanyorientationandanylocationinspace.(The

planecanmisssomeoralldatapoints.)Theresultofthealgorithmshouldbeamodel

ofthedensityofthescannedobjectovertheselectedplane.

ProblemBGradeInflation

Background

SomecollegeadministratorsareconcernedaboutthegradingatABetterClass(ABC)

college.Onaverage,thefacultyatABChavebeengivingouthighgrades(theaverage

gradenowgivenoutisanA-),anditisimpossibletodistinguishbetweenthegoodand

mediocrestudents.Thetermsofaverygenerousscholarshiponlyallowthetop10%of

thestudentstobefunded,soaclassrankingisrequired.

Thedeanhadthethoughtofcomparingeachstudenttotheotherstudentsineach

class,andusingthisinformationtobuilduparanking.Forexample,ifastudent

obtainsanAinaclassinwhichallstudentsobtainanA,thenthisstudentisonly

“average"inthisclass.Ontheotherhand,ifastudentobtaintheonlyAinaclass,

thenthatstudentisclearly"aboveaverage”.Combininginformationfromseveral

classesmightallowstudentstobeplacedindeciles(top10%,next10%,ect.)acrossthe

college.

Problem

Assumingthatthegradesgivenoutare(A+,A-,B+,B-,...)canthedean'sideabemadeto

work?

Assumingthatthegradesgivenoutareonly(A,B,C,...)canthedean'sideabemadeto

work?

Cananyotherschemesproduceadesiredranking?

Aconcernisthatthegradeinasingleclasscouldchangemanystudent，sdeciles.Is

thispossible?

DataSets

Teamsshoulddesigndatasetstotestanddemonstratetheiralgorithms.Teamsshould

characterizedatasetsthatlimittheeffectivenessoftheiralgorithms.

MathematicalContestinModeling1999Problems

ProblemA-DeepImpact

Forsometime,theNationalAeronauticsandSpaceAdministration(NASA)hasbeen

consideringtheconsequencesofalargeasteroidimpactontheearth.

Aspartofthiseffort,yourteamhasbeenaskedtoconsidertheeffectsofsuchan

impactweretheasteroidtolandinAntarctica.Thereareconcernsthatanimpactthere

couldhaveconsiderablydifferentconsequencesthanonestrikingelsewhereonthe

planet.

Youaretoassumethatanasteroidisontheorderof1000mindiameter,andthatit

strikestheAntarcticcontinentdirectlyattheSouthPole.

Yourteamhasbeenaskedtoprovideanassessmentoftheimpactofsuchanasteroid.

Inparticular,NASAwouldlikeanestimateoftheamountandlocationoflikelyhuman

casualtiesfromthisimpact,anestimateofthedamagedonetothefoodproduction

regionsintheoceansofthesouthernhemisphere,andanestimateofpossiblecoastal

floodingcausedbylarge-scalemeltingoftheAntarcticpolaricesheet.

ProblemB-UnlawfulAssembly

Manypublicfacilitieshavesignsinroomsusedforpublicgatheringswhichstatethat

itis“unlawful"fortheroomstobeoccupiedbymorethanaspecifiednumberofpeople.

Presumably,thisnumberisbasedonthespeedwithwhichpeopleintheroomcouldbe

evacuatedfromtheroom，sexitsincaseofanemergency.Similarly,elevatorsandother

facilitiesoftenhave“maximumcapacities"posted.

Developamathematicalmodelfordecidingwhatnumbertopostonsuchasignasbeing

the“l(fā)awfulcapacity”.Aspartofyoursolutiondiscusscriteria,otherthanpublic

safetyinthecaseofafireorotheremergency,thatmightgovernthenumberofpeople

considered〃unlawful〃tooccupytheroom(orspace).Also,forthemodelthatyou

construct,considerthedifferencesbetweenaroomwithmovablefurnituresuchasa

cafeteria(withtablesandchairs),agymnasium,apublicswimmingpool,andalecture

hallwithapatternofrowsandaisles.Youmaywishtocompareandcontrastwhatmight

bedoneforavarietyofdifferentenvironments:elevator,lecturehall,swimmingpool,

cafeteria,orgymnasium.Gatheringssuchasrockconcertsandsoccertournamentsmay

presentspecialconditions.

Applyyourmode1tooneormorepublicfacilitiesatyourinstitution(orneighboring

town).Compareyourresultswiththestatedcapacity,ifoneisposted.Ifused,your

modelislikelytobechallengedbypartieswithinterestsinincreasingthecapacity.

Writeanarticleforthelocalnewspaperdefendingyouranalysis.

2000MathematicalContestinModeling

ProblemAAirtrafficControl

DedicatedtothememoryofDr.RobertMachol,formerchiefscientistoftheFederalAviation

AgencyToimprovesafetyandreduceairtrafficcontrollerworkload,theFederalAviation

Agency(FAA)isconsideringaddingsoftwaretotheairtrafficcontrolsystemthatwould

automaticallydetectpotentialaircraftflightpathconflictsandalertthecontroller.

Tothatend,ananalystattheFAAhasposedthefollowingproblems.

RequirementA:Giventwoairplanesflyinginspace,whenshouldtheairtrafficcontroller

considertheobjectstobetoocloseandtorequireintervention?

RequirementB:Anairspacesectoristhesectionofthree-dimensionalairspacethatone

airtrafficcontrollercontrols.Givenanyairspacesector,howdowemeasurehowcomplex

itisfromanairtrafficworkloadperspective?Towhatextentiscomplexitydetermined

bythenumberofaircraftsimultaneouslypassingthroughthatsector(1)atanyoneinstant?

(2)duringanygivenintervaloftime?(3)duringaparticulartimeofday?Howdoesthe

numberofpotentialconflictsarisingduringthoseperiodsaffectcomplexity?

Doesthepresenceofadditionalsoftwaretoolstoautomaticallypredictconf1ictsandalert

thecontrollerreduceoraddtothiscomplexity?

Inadditiontotheguidelinesforyourreport,writeasummary(nomorethantwopages)

thattheFAAanalystcanpresenttoJaneGarvey,theFAAAdministrator,todefendyour

conclusions.

ProblemBRadioChannelAssignments

Weseektomodeltheassignmentofradiochannelstoasymmetricnetworkoftransmitter

locationsoveralargeplanararea,soastoavoidinterference.Onebasicapproachisto

partitiontheregionintoregularhexagonsinagrid(honeycomb-style),asshowninFigure

1,whereatransmitterislocatedatthecenterofeachhexagon.

Figure1

Anintervalofthefrequencyspectrumistobeallottedfortransmitterfrequencies.The

intervalwillbedividedintoregularlyspacedchannels,whichwerepresentbyintegers

1,2,3,….Eachtransmitterwillbeassignedonepositiveintegerchannel.Thesame

channelcanbeusedatmanylocations,providedthatinterferencefromnearbytransmitters

isavoided.Ourgoalistominimizethewidthoftheintervalinthefrequencyspectrum

thatisneededtoassignchannelssubjecttosomeconstraints.Thisisachievedwiththe

conceptofaspan.Thespanistheminimum,overal1assignmentssatisfyingtheconstraints,

ofthelargestchannelusedatanylocation.Itisnotrequiredthateverychannelsmaller

thanthespanbeusedinanassignmentthatattainsthespan.

Letsbethelengthofasideofoneofthehexagons.Weconcentrateonthecasethatthere

aretwolevelsofinterference.

RequirementA:Thereareseveralconstraintsonfrequencyassignments.First,notwo

transmitterswithindistanceofeachothercanbegiventhesamechannel.Second,dueto

spectralspreading,transmitterswithindistance2sofeachothermustnotbegiventhe

sameoradjacentchannels:Theirchannelsmustdifferbyatleast2.Undertheseconstraints,

whatcanwesayaboutthespanin,

RequirementB:RepeatRequirementA,assumingthegridintheexamplespreadsarbitrari1y

farinalldirections.

RequirementC:RepeatRequirementsAandB,exceptassumenowmoregenerallythatchannels

fortransmitterswithindistancedifferbyatleastsomegivenintegerk,whilethoseat

distanceatmostmuststilldifferbyatleastone.Whatcanwesayaboutthespanandabout

efficientstrategiesfordesigningassignments,asafunctionofk?

RequirementD:Considergeneralizationsoftheproblem,suchasseverallevelsof

interferenceorirregulartransmitterplacements.Whatotherfactorsmaybeimportantto

consider?

RequirementE:Writeanarticle(nomorethan2pages)forthelocalnewspaperexplaining

yourfindings.

2001MathematicalContestinModeling(MCM)

ProblemA:ChoosingaBicycleWheel

Cyclistshavedifferenttypesofwheelstheycanuseontheirbicycles.Thetwobasictypes

ofwheelsarethoseconstructedusingwirespokesandthoseconstructedofasoliddisk

(seeFigure1)Thespokedwheelsarelighter,butthesolidwheelsaremoreaerodynamic.

Asolidwheelisneverusedonthefrontforaroadracebutcanbeusedontherearof

thebike.Professionalcyclistslookataracecourseandmakeaneducatedguessastowhat

kindofwhee1sshouldbeused.Thedecisionisbasedonthenumberandsteepnessofthe

hiIls,theweather,windspeed,thecoinpetition,andotherconsiderations.Thedirector

sportifofyourfavoriteteamwouldliketohaveabettersysteminplaceandhas

askedyourteamforinformationtohelpdeterminewhatkindofwheelshouldbeusedfor

agivencourse.

Figure1:Asolidwheelisshownontheleftandaspokedwheelisshownontheright.

Thedirectorsportifneedsspecificinformationtohelpmakeadecisionandhasaskedyour

teamtoaccomplishthetaskslistedbelow.Foreachofthetasksassumethatthesamespoked

wheelwillalwaysbeusedonthefrontbutthereisachoiceofwheelsfortherear.

Task1.Provideatableivingthewindpeedatwhichthepowerrequiredforasolid

rearwheelislessthanforaspokedrearwheel.Thetableshouldincludethewindspeeds

fordifferentroadgradesstartingfromzeropercenttotenpercentinonepercentincrements.

(Roadgradeisdefinedtobetheratioofthetotalriseofahi11dividedbythelength

oftheroad.Ifthehillisviewedasatriangle,thegradeisthesineoftheangleat

thebottomofthehill.)Ariderstartsatthebottomofthehillataspeedof45kph,

andthedecelerationoftheriderisproportionaltotheroadgrade.Ariderwilllose

about8kphforafivepercentgradeover100meters.

Task2.Provideanexampleofhowthetablecouldbeusedforaspecifictimetrialcourse

Task3.Determineifthetableisanadequatemeansfordecidingonthewheelconfiguration

andofferothersuggestionsastohowtomakethisdecision.

ProblemB:EscapingaHurricane'sWrath(AnIllWind...)

EvacuatingthecoastofSouthCarolinaaheadofthepredictedlandfallofHurricaneFloyd

in1999ledtoamonumentaltrafficjam.TrafficslowedtoastandstillonInterstate1-26,

whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafe

havenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrive

tookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.Fortunately,

Floydturnednorthandsparedthestatethistime,butthepublicoutcryisforcingstate

officialstofindwaystoavoidarepeatofthistrafficnightmare.

Theprincipalproposalputforthtodealwiththisproblemisthereversaloftrafficon

1-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfrom

CharlestontoColumbia.Planstocarrythisouthavebeenprepared(andpostedontheWeb)

bytheSouthCarolinaEmergencyPreparednessDivision.Trafficreversalonprincipalroads

leadinginlandfromMyrtleBeachandHiltonHeadisalsoplanned.

AsimplifiedmapofSouthCarolinaisshown.Charlestonhasapproximately500,000people,

MyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongthe

restofthecoastalstrip.(Moreaccuratedata,ifsought,arewidelyavailable.)

Theinterstateshavetwolanesoftrafficineachdirectionexceptinthemetropolitanareas

wheretheyhavethree.Columbia,anothermetroareaofaround500,000people,doesnothave

sufficienthotelspacetoaccommodatetheevacuees(includingsomecomingfromfarthernorth

byotherroutes),sosometrafficcontinuesoutboundon1-26towardsSpartanburg;on1-77

northtoCharlotte;andon1-20easttoAtlanta.In1999,trafficleavingColumbiagoing

northwestwasmovingonlyveryslowly.Constructamodelfortheproblemtoinvestigate

whatstrategiesmayreducethecongestionobservedin1999.Herearethequestionsthat

needtobeaddressed:

1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesof1-26into

twolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-way

traffic,significantlyimproveevacuationtrafficf1ow?

2.In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.

Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggersthe

evacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternof

howhurricanesaffectthecoast?

3.SeveralsmallerhighwaysbesidesI_26extendinlandfromthecoast.Underwhatconditions

woulditimproveevacuationflowtoturnaroundtrafficonthese?

4.Whateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersin

Columbia,toreducethetrafficleavingColumbia?

5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotor

homes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictions

onvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?

6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,

whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameup

1-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuation

trafficflow?Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.

Required:Prepareashortnewspaperarticle,nottoexceedtwopages,explainingthe

resultsandconclusionsofyourstudytothepublic.

問題A:選擇自行車車輪

騎自行車的人有幾種不同類型的車輪可以用在他們的自行車上。其中兩種基本類型是使用

了輻條的車輪和使用實心盤的車輪（見圖1）。使用輻條的車輪比較輕，而實心輪子就比較符合

空氣動力學(xué)。在公路自行車比賽中，一個實心的輪子從來不會被用于做前輪，而可以做后輪。

專業(yè)自行車（賽車）手們在賽車場上觀察，并做有根據(jù)的推測，決定應(yīng)該使用什么類型的

輪子。所作的決定是基于小山波的數(shù)目和險峻程度、天氣、風(fēng)速、競爭對手和其他考慮因素。

你喜愛的隊的教練希望用一個更好的系統(tǒng)代替（原有的），并已經(jīng)咨詢過你的隊伍，以幫助決

定在某種特定情況下應(yīng)該使用那種輪子。

圖1:左邊的是實心輪子，右邊的是裝有條幅的輪子

教練需要細節(jié)信息去幫助做決定，并已經(jīng)要求你的隊伍去完成下面列出的任務(wù)。每個

任務(wù)假定同樣的裝有輻條的車輪總是裝在前面，而后輪則可以選擇。

?任務(wù)1.給出一個表格，在不同風(fēng)速下一個實心輪與條幅輪所需能量哪個較多。表格應(yīng)該包括

一個對應(yīng)不同道路等級（從0%到10%,增量為1%）的風(fēng)速（道路等級定義為斜坡高度與長度之比,

如果斜坡視為一個三角，則等級就是山腳處的斜角正弦）。一個車手從山腳開始，初速為45kph,

而該車手的減速率與道路等級成正比，例如，在5%等級道路上大約每100米減速8kph。

?任務(wù)2.給出個例子如何對一個指定的時間試驗應(yīng)用上述表格。

?任務(wù)3.判定該表格是否是一個適當(dāng)?shù)姆椒ㄈQ定車輪的配置并提供其他解決該決策的方案。

問題B:逃命吧！〃颶風(fēng)的憤怒〃（一種III級的風(fēng)）來了!

1999年在預(yù)報過的颶風(fēng)"佛洛伊德”登陸之前，美國南卡羅萊納州海岸的疏散工作導(dǎo)致了一

場極端的交通擠塞。在州際126號公路上的交通近乎于停滯，該公路是從查爾斯頓（美國西弗吉

尼亞州首府）到哥倫比亞（該州中部）的相對安全的避難所的首要路線。通常是輕松的兩小時

車程結(jié)果變成了18小時才能完成。很多車輛在路上完全耗光了汽油。不幸中的萬幸是，這次“

佛洛伊德”颶風(fēng)轉(zhuǎn)向北方并減弱了。但是，公眾的大聲疾呼令該州官員不得不尋找方法去