2013年美赛优秀论文A题O奖

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1、SummaryManyconventionalrectangularbakingpanshaveaproblemthattheybakethecornersofthebatterfasterthantherestofthepan.Circularbakingpanseliminatethisproblem,buttakeupmorespaceintheoven.Weproposeasolutionthatgivenweightsforbakingconsistencyandspaceefficiencybasedonth

2、eimportanceofeachwillprovidetheoptimalbakingpanshape.WehavecomeupwithanalgorithmforsortingthepansofareaAeffectivelyandformedamodelthatdescribestheheatflowintothepanaswellasthebakingmixitself.Themodelwedevelopedwasbasedoffreal-lifedatathatwegatheredbybakingbrownie

3、susingcarefulmeasurement.Wedeterminedseveralphysicalproperties,suchasspecificheat,density,andheattransfercoefficientsforbrowniesdirectlythroughexperimentation.Wecomparedourresultswithpreviousstudiesonthepropertiesofcake-likefoodsandfoundthattheyweresimilar.Thenwe

4、ranthemodelusingthesecoefficientsandavarietyofdifferentplausiblebakingparameters,includingtemperatureandbakingtime.Wealsodevelopedatechniqueforoptimallyarrangingpolygonalshapedpansintheovenforanylengthto-widthratiotomaximizethenumberofpansthatcanfit.Forpolygons,w

5、econsideredbothlatticearrangementsandwhereapplicable,tessellatingpatterns.Thealgorithmforpackingalsotakesintoaccounttheneedforairflowbetweenthepansforoverallevencooking,whichmeansthatthepansarenotpackeddirectlynexttoeachothercausingunevenheatingonthepanslocatedne

6、arthesidesoftheoven.Theoverallefficiencyofapanshapewascalculatedusingthefollowingequation(wherepisaweightingfactor):Efftotal=p·Effspace+(1−p)·EffheatingUsingthealgorithmwefoundthatthetypeofpolygonneededtoefficientlyusetheovenspacewhileheatingthepanperimeterwasgre

7、atlydependentonthedimensionsoftheovenforweightingvaluesbetween0.2-1.0.(Below0.2,thecircularwww.madio.net数学中国panisveryheavilyfavoredregardlessoftheovendimensions.)Wealsofoundthataddingorsubtractingalittlebitfromthewidthorlengthoftheovencanhaveadrasticeffectonthesp

8、acingefficiencyratio.Whenwehavemaximallyhighefficiency,wecallthewidthandlengthofthepan”magicnumbers.”Forexample,iftheovendimensions(widthbylength)are30”x40”,wi