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Brewing optimal coffee
Article in European Journal of Physics · November 2020 DOI: 10.1088/1361-6404/abc97d
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2 authors, including: Ann Smith University of Huddersfield
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BrewingOptimalCoffee
A.SmithandW.T.Lee∗ SchoolofComputingandEngineering,UniversityofHuddersfield (Dated:February3,2021)
Coffeeisabeverageenjoyedworldwideandanactiveareaofcurrentresearch.Thebrewing ofcoffeealsopresentsanopportunitytodemonstratethemodel,simulateandoptimiseparadigm inaction.Thisisdoneinthecontextofbrewingcafeti`erecoffee.Usingapublishedmodeland experimentaldataweshowthatanexistingmodelcanbeparameterisedtosimulatetheresultsof thatexperiment.Fromhereanoptimisationstepcanbeappliedinwhichtheexperimentalrecipe canbechangedtoimprovethequalityofthecoffeeproducedintheexperiment.Thisactivityis implementedintheNUMBASplatform.
I.INTRODUCTION
Wereportanactivitywehavedevelopedforopendays inwhichstudentsareguidedthroughthestepsofthe modelling,simulationandoptimisationparadigminthe contextofbrewingcoffee.Thisallowsstudentstoseein averyshorttimehowsuchmodelsareusedinrealworld applicationswheretheaimisnotjusttounderstandhow asystemoperatesbutalsotoadjustittomakeitwork better.Usingcoffeeasanexamplehasanumberofadvantagesforthis.Firstlyitisfamiliartostudentsand cansafelybeconsumedbythem,sotheycanexperience throughthesenseoftastehowmodellinghasimproved things.Secondly,thefactthatasfamiliarasubjectas coffeebrewingisstillanareaofactiveresearchallowsus tohighlighttostudentsjusthowbroadtheopportunities availabletoapplymodellingskillsare.Finally,thereisa publishedmodelanddatasetandaquantitativedefinitionof‘good’coffee.
Legendhasitthatthestimulatingpropertiesofcoffeewerediscoveredbyagoatherdwhonoticedthathis chargesbecameunusuallyspiritedafterconsumingthe cherriesofacertainplant.1 Thisofcoursewasthecoffee plantwhichcomesin robusta and arabica versions.Todayworldcoffeeconsumptionstandsatover160million 60kgbagsofcoffeeayear.2
Inchangingfromthegoatspreferredmethodofconsumingcoffeetobrewingcafeti`erecoffee,oneoftheways humansprefertotakethisstimulant,anumberofadditionalstagesareneeded.Thecherriesareharvestedand theirseeds,colloquiallyandbelowcalledcoffeebeans,are extracted.Intheirnaturalstatecoffeebeansareagreen color,buttheyareroastedtoadarkbrowncolor;this stageintroducessomeofthemorecomplexflavoursinto thecoffee.Thebeansarenextgroundtoasmallsize. Atypicalsizedistributionisbimodalwithlargeandfine grainsizes.Thishasastrongeffectonextractionalthoughwewillignorethisinoursimplermodels.Finally thegroundcoffeeissteepedinboilingwater,transferring someofthesolublecontentofthecoffeetothewater.The resultingsolutionisthedrinkknownascoffee.
Chemicallycoffeeisincrediblycomplex,consistingof nearly2000chemicals,3 asimilarlevelofchemicalcomplexitytowine.Thesearesometimesgroupedintothree
categoriesroughlybymolecularweight.Thelightest groupsarevolatilearomatics,largelyresponsibleforthe smellofcoffee.Themiddlecategoryarecomplexsugarsandalkaloidssuchascaffeine.Thefinalandheaviest componentsarelargecomplexsugarsresponsibleforthe bittertasteofcoffee.Thislattergroupmustbepresent insmallamountstogivebrewedcoffeesomecharacter, butinlargeamountsgiveitanoverwhelminglybitter taste.
Inthispaperwefocusonbrewingofcoffeeina cafeti´ere,howevertherearenumerousothermethodsof brewingcoffee.Instantcoffeeismadebybrewingand thenfreezedryingtheresultingsolution,allowingthe coffeetobereconstitutedbytheadditionofhotwater. Inespressocoffeemakingwaterisforcedthroughfinely ground,denselypackedcoffeegrainsathighpressures(6 to8atmospherestypically).Themostinterestingwayof makingcoffeefromaphysicsperspectiveisprobablyby themokapotwhichexploitssimilarphysicstoageyser toforceboilingwaterthroughacoffeebed.Manyother waysofbrewingcoffeeareused.
Anumberofattemptshavebeenmadetodevelop mathematicalmodelsofthebrewingofcoffeebyvariousmethods.Aninstrumentedmokapotwasusedto developamodelofthethermodynamicsandfluidmechanicsofthatbrewingmethod,howeverthephysics ofextractionwerenotconsidered.4,5 Partialdifferential equationmodelsofextractioninespressoweredeveloped byFasanoetal.6 Thesemodelsweresetuptoconsider themulticomponentnatureofcoffeebutasthemainaim oftheauthorswastoinvestigatethepropertiesofthe equationsthemselves,e.g.existenceanduniquenessof solutions,theydidnotattempttoparameterisethem.A multiscalemodelofcoffeebrewingwasdevelopedbyMoroneyetal.leveragingmodelsofgroundwaterflowand focussedonfiltercoffeeextraction.7,8 AnalternativemultiscalemodelbasedonLiionbatterieshasbeenapplied toespressobrewing.9 Withtheexceptionofthemodels developedbyFasanoetal.,standardpracticeinthisarea istotreat‘coffee’asasinglecomponentsubstanceand describeitintermsofamassconcentration.
Thebrewingofgoodcoffeeisanartratherthana science,howeversomeroughguidelinesonhowtobrew coffeeexistinthegoldencupformuladevelopedbythe SpecialityCoffeeAssociation.Thisguidancerefersto
FIG.1.Thecoffeebrewingcontrolchart,developedbythe SpecialityCoffeeAssociation.Thisgivesaroughguideto makingnon-espressocoffee.
brewingmethodsexcludingespressoandisencodedin thecoffeebrewingcontrolchart.Thisdescribesacup ofcoffeeintermsoftwovariables:extractionyieldand strength.Astheseformthe x-and y-axesofthecoffee controlchartrespectively,wedenotetheseby x and y Extractionyield, x,isthemasspercentageofthecoffee groundsthatgoesintosolution.Strength, y istheconcentrationofthecoffeesolutionexpressedasamasspercentage,i.e.thefractionbyweightofacupofcoffeethat isdissolvedcoffeesolids.Optimalrangesofthesequantitiesweredeterminedthroughtastetests.10 Anexample coffeecontrolchartappropriateforfilterandcafeti`ere coffeeisshowninFig.1.
Thetwokeyassumptionsthatgointoacoffeecontrol chartarethatextractionisuniformthroughoutthecoffee bedandtheconcentrationof‘coffee’canbedescribedin termsofasinglevariable.Infactextractionyielddoes takeintoaccountthemulticomponentnatureofcoffee. Asmallextractionyieldindicatesthatthechemistryof thedissolvedcoffeeisdominatedbylowmolecularweight volatilecompounds,whereaslargeextractionyieldsindicateacupofcoffeewhoseflavourisdominatedbyhigh molecularweightbittertastingchemicals.Someeffort hasbeenmadetouseacoffeecontrolcharttoindicate therangeofextractionyieldsoccurring,bothasafunctionofgrainsizeandpositionwithinthecoffeebed.11
Ifthetwoaforementionedassumptionsareaccepted then,byconservationofmass,itiseasytoseethatany coffeebrewingprocesswillbedescribedbyastraightline. Let Mw and m be,respectively,themassofwaterand groundcoffeeused.Ifanamountofcoffee∆m goesinto solutionthen(assuming∆m Mw whichistrueforfiltercoffee)extractionyield, x,isgivenby x =∆m/m and strength, y,isgivenby y =∆m/Mw.Eliminating∆m fromtheequationsfor x and y gives y =(m/Mw)x.Thus anybrewingprocessisastraightlinepassingthroughthe
TABLEI.Coffeeconcentrationdata.7
originwithagradientequaltothegroundcoffeetowaterratio.Thegoldencupformularecommendsacoffee groundstowaterratioof30gcoffeeperkgorlitreof water.Ofcoursethisinitselfdoesnotguaranteegood tastingcoffeeoreventhattheeventualcupwillbein theoptimalregionofthecoffeecontrolchart.Tastewill alsodependonthebrewingtime,controlleddirectlyby theoperatorinthecaseofacafeti`ere,butbyarangeof morecomplexfactorssuchasgrindsizedistributionin thecaseoffiltercoffeeoramokapot.
Apublishedexperiment,7 gaveaninsightintobrewing dynamicsunderconditionssimilarto,butnotexactly thesameas,cafeti`erebrewing.Inthisexperimentcoffee groundsandwaterweremixedataratioof60gcoffee groundsto1litreofwaterandstirredcontinuously.This experimentissimilartocafeti`erebrewingbut,duetothe stirring,theextractionkineticsarelikelytobefaster. Alsothefinalstageinwhichthefilterispusheddownto trapthegrainsatthebaseofthecafeti`ereisnotincluded. Atregularintervalsduringtheexperimentasmallsample oftheliquidwasextractedandthemassconcentration ofcoffee, c,wasmeasuredbyrefractometry(forsmall concentrationsdissolvedcoffeeaffectstherefractiveindex ofwaterinaknownway).Theconcentrationisthemass ofcoffeeinsolutionperunitvolumeofwater: c =∆m/V where V = Mw/ρw.(Wehaveassumedthat c issmall sothatthevolumeoftheliquidisnotchangedbythe dissolvedcoffee.)ThesedataaretabulatedinTableI andplottedinFig.2.
Fromtheconcentration,theextractionyield, x,and strength, y,canbecalculatedasshowninTableII.Fig.3 andFig.4show x and y plottedasfunctionsoftime.
Wecanalsoplotthe x and y dataonthecoffeecontrol chart.ThisisshowninFig.5.Theplotshowsthatthis experimentwillproducecoffeethatisfartoostrongto bepalatable.Inthecontextoftheexperimentthisis reasonable:itisaimedatobtainingscientificdatanot makinggoodcoffee.However,wecanaskthequestion: iftheaimwastocreategoodcoffeehowshouldtheexperimentbemodified?Specifically:Whatwatertocoffee ratioshouldbeusedandforhowlongshouldthecoffee bebrewed?
Therestofthepaperisorganisedasfollows.InSectionIIwereviewthestateoftheartinmodellingthe physicsandchemistryofcoffeeextraction,includingan
TABLEII.Extractionyieldandstrengthcorrespondingto concentrationsgiveninTableI.
applicationofthatmodellingframeworktotheexperimentdescribedabove,anddevelopandparameterisea setofequationsdescribingthissystem.Wealsoinvestigatewhatthesemodelstellusaboutthewaytooptimise coffeebrewingwiththisexperiment.InSectionIIIwe discusshowthesestepscanbesimplifiedtoproducean activitystudentscanworkthroughinarelativelyshort session,thisrequiressimplifyingthetheoryandparame-
FIG.5.Experimentaldataandbestfitlineplottedonthe coffeecontrolchart.Thisshowsthattheexperimentwillbrew coffeemuchtoostrongtobeoptimalforhumanconsumption.
terisationofthemodelconsiderably.Wedescribeourimplementationofanelectronicworksheetguidingstudents throughthecalculation,anddiscusstheerrorsourresults havegenerated.InSectionIVwediscusspotentialextensionsofthismodel.Theactivityasdescribedisaimed atstudentsenteringuniversity,moreadvancedstudents withagraspofcurvefittinganddifferentialequations caninvestigatethissysteminalotmoredepth.Finally, wegiveconclusionsinSectionV.
II.THEORETICALBACKGROUND
BuildingonpreviousworkbyFasano6 andmodelsof chemicalextractioningroundwaterflow,12 Moroneyet al.developedasetofequationsthatcandescribecoffeebrewinginboththefiltercoffeegeometryandinthe cafeti`ereexperimentdescribedabove.Theseequations trackcoffeeinsolidanddissolvedformswithinlargeand smallcoffeegrainsandindissolvedforminporesbetween
FIG.2.Brewingdata
FIG.3.Plotofextractionyield, x againsttime.
FIG.4.Plotofcoffeestrength, y againsttime.
grains.Moroneyetal.developedtheseequationsforfiltercoffeeinitiallyandthen,byneglectingconvectionand diffusionterms,appliedthemtothecafeti`ereexperiment.
Theseequationscanbereformulatedasacompartment modeltrackingmassesofsolublecoffeepresentinlarge grains, m1,smallgrains, m2,anddissolvedinthewater mw
levelorprovidingextensionquestionstomoreablestudents.
Tomakeitpossibletoletstudentscompletetheactivitywithintherelativelyshorttimeavailableseveral simplificationsareused
• Coffeeextractionisconsideredtobeasingle timestepprocesssothattheextractionequations become x = x0(1 exp( t/τ )),and y havinga similarstructure.Thisisequivalenttoconsideringthatcoffeegrainsaremonodisperseratherthan separatedintosmallandlargegrains.
m1 = m10 at t =0(4)
m2 = m20 at t =0(5)
mw =0at t =0(6)
Onesimplifyingassumptionwehavemadeistoassume thatthecoffeeisveryweaksothecoffeealreadyinsolutiondoesnothinderextraction.Theseequationshave solutions
• Studentsarenotexpectedtosolvedifferentialequations,solutionsareprovided.Furthermore,as notedabove,weneglecttheeffectofhinderingof coffeeextractionbycoffeealreadyinsolution,i.e. wewillassumethatwhenwechangethewaterto coffeeratiotheequationfor x anditsparameters willnotchange.
If m isthetotal(initial)massofgroundcoffeeand Mw istheinitialmassofwater,correspondingtoavolumeof water Vw = Mw/ρw then x = mw m (10) y = mw Mw (11)
c = mw Vw (12)
where x isextractionyield, y isstrengthand c isconcentration.Notethat m = m10 + m20 becausenotallthe massofthegrainsissolublewhereas m10 and m20 are solublemasses.
III.ACTIVITY
Anopendayactivityaimedatvisitingschoolstudents consideringattendingouruniversitywasdevelopedbased onthisdata.Inthisactivityweaskstudentshowtheexperimentdescribedaboveshouldbemodifiedtoproduce drinkablecoffee.Thisworksoutasaskingtwoquestions: whatisthecoffeetowaterratiothatshouldbeusedand howlongshouldthebrewinghavebeencarriedoutfor? Toaddressthesequestionsatanaccessiblelevelandin arelativelyshortamountoftimewesimplifythetheoryconsiderably.Inthenextsectionwediscusspossible extensionstothetheorythatcouldbeofinteresttoeducatorswishingtoadapttheactivityatamoreadvanced
Thestudentsareguidedthroughthefollowingsteps. Firstlytheyareintroducedtothecoffeecontrolchart andaskedtointerpretsomedata.Theyaregiven x and y coordinatesandaskedtousethecoffeecontrolchartto interpretthesecoordinatesintermsofthetasteofcoffee. Thentheyaregivensomedataforinitialandfinalmass ofcoffeegrainsandmassofwaterusedandaskedto calculate x and y andinterprettheseasthetasteofthe resultingcoffee.
InthenextstepstudentsareintroducedtotheFrench pressdataandtheaimoftheactivity.TheFrenchpress dataispresentedinitsoriginalformasconcentration data.Studentsareaskedtoconvertsomerandomlychosenpointsinto x and y coordinates.Followingthisstudentsaregivenacompletedatasetandtheequations for x and y (simplifiedtoasingletimescaleasdiscussed above).Studentsarethenaskedtoestimatetheparameters x0 and τ byusingthefinaltimepointasanestimate of x0 andthefirstnonzerotimepointtofind τ .These resultintheestimates x0 =30 25%and τ =22 25s.The resultingfittothedataisshowninFig.6.
Usingthisparameterisation,studentsthensolvethe equationfor x attheoptimalextractionlevelof20% fortime, t =24.08s.Theyarethenaskedtocalculate thecorrespondingvalueof y (2 4%).Thenextstepis toconsiderwhatthesecoordinatesmeanintermsofthe tasteofcoffeewithreferencetothecoffeebrewingcontrol chart.Studentsshouldcorrectlydeducethatthiscoffee isfartoostrong:somuchsothatitwillnotevenappear onthechartasitisusuallyplotted.
Havingcompletedthesimulationstepstudentsare nowaskedtomoveontotheoptimisationstep.Making theassumptionthattheparameterisationchosenforthe x(t)equationwillnotchangeifthewatertocoffeeratio changes,studentsareaskedtocalculatethewatertocoffeeratiothatwouldresultinoptimalcoffee.(Thisshould comeouttobeabout31.25g.)Theeffectofchangingthe massofcoffeeonthestrengthisshowninFig.7.
FIG.6.Algebraicfittoextractionyield, x,data.Thefirst andlastpointsareusedtoestimatethetimescale τ andsaturationpoint x0
typicallyagreesthattheoptimalcoffeeisbest.
Severalsimplifyingassumptionswereintroducedwhich affecttheresultstovariousextents.Fromthedatait seemsreasonablethatthelastdatapointdoesindeedcorrespondtothesaturationlevelsothisisnotexpectedto introduceanyerrors.Theassumptionthatthetimescale τ canbecalculatedfromthefirstdatapointisonlyas accurateastheassumptionthatthesingleexponential processaccuratelydescribestheexperimentaldata:as wehaveseenabovethisisnottrue.MoreaccuratefittingproceduresarediscussedinsectionIV.Notehoweverthatthiswillnotaffecttheoptimalratioofcoffee towaterasthisisindependentoftheexactformofthe extractioncurveused.
ThisactivitywasimplementedwithintheNUMBAS platform.13,14 Aversionoftheactivitythatcanberun ona(networked)webbrowserisavailablefromthislink. Editableversionsofthequestionscanbeaccessedonthe NUMBASwebsiteandarereleasedunderacreativecommonslicenceallowinganymodifications.TheNUMBAS platformisaninteractiveenvironmentforimplementing mathematicalquestionsandacceptsalgebraicaswellas numericalanswerstoquestions.Ithastheabilitytorandomisenumericalparameterswithinaquestion.Italso facilitatesscaffoldingbyincorporatingundisclosedsupportinginformationwhichisavailableonrequest,thus guidinganystrugglingstudentstowardsasolution.The stagesoftheactivityareasdescribedabove.
IV.EXTENSIONS
FIG.7.Adjustingthecoffeetowaterratiotoimprovetaste. Strengthisshownplottedagainsttimewiththetimeat whichtheextractionyieldisoptimalshownbyaverticalline. Theoptimalstrengthisshowningrey.Thebluelineshows strengthasafunctionoftimefortheoriginalmassofcoffee. Theredlineshowsthatifthemassofcoffeeusedischanged to31.25goptimalcoffeecanbebrewed.
Withthemathematicalsideofthingscompletewenow movetoanexperimentaltest.Usingsomescales,akettle,somegroundcoffeeandtwocafeti`eres,twodifferent coffeesarebrewed.(Astudentcanbeaskedtoconfirm themassesarecorrectortoweighoutthegroundcoffee, butitisprobablysafesttoletastaffmemberpourthe hotwater!)Sincecafeti`eresarenotcontinuouslystirred, thebrewingtimestudentshavecalculatedwillnotbeaccurateanditisbettertowaittherecommendedthree minutesforthecoffeetobrew.Duringthisperiodof brewingyoucangooversomeoftheideasforextensionsdiscussedbelowinsectionIV.Avolunteeristhen soughttotastethetoostrongcoffeeandthe‘optimal’ coffee.Thevolunteer(onopendaysusuallyaparent)
Thisactivityisconsiderablyshortenedtoallowitto berunwithinanopendaysession.Numerousextensionsarepossible.Theseincludeasimplercalculationof thecoffeetowaterratiowithouttheintermediatestepof calculatingthetime,deriving x(t)and y(t)fromadifferentialequation,animprovedparameterisationofthe modelfromleastsquaresfittingusingtheexcelsolveror anotherminimisationtechnique;andvisualisationofthe datae.g.inanexcelchartorbyageogebrawidget.
a.Asimpleratioderivation. Thecalculationofthe m/Mw ratiopresentedaboveisslightlytooinvolvedif onlytheoptimalbrewingratioisneededandnotthe brewingtime.Thiscanbedirectlycalculatedfromthe relationshipbetween x and y withoutmakinganyassumptionsaboutthebrewingkineticsiftheyarenotof interest.Sincecoffeesolidsareconservedwealwayshave y = mx/Mw sobypickingoptimalvaluesof x and y from thecoffeecontrolchartwecaninstantlydeterminethe correctvalueofthebrewingratio m/Mw .Ifonestudent finishedthematerialearlythisisagoodbonusquestion toask.Alternativelyitcanbediscussedintheinterval whilethecoffeebrews.
b.DerivingtheEquations Intheactivitytheequationsfor x(t)and y(t)areproduceddirectly.However thesecanbederived.Thefullequationsarerelatively complexbutasimplifiedmodelfor x(t)astheresultof
aseparabledifferentialequationcanbederivediftheassumptionofasinglekinetictimescaleismaintained.In thiscaseitisassumedthattherateofdissolution,i.e.increaseof x,isproportionaltothesolublecontentofthe coffeegrains x0 x where x0 isthemaximumpossible valueof x thesolublecontentofthecoffeegrains.This leadsto˙x ∝ (x0 x)andintroducingatimescale, τ ,as aconstantofproportionalityleadsto
dx dt = x0 x τ (13) x =0at t =0(14)
whichcanbeintegratedtogive
x = x0 1 e t/τ , (15)
y = mx0 Mw 1 e t/τ (16)
(Wheretheequationfor y isdeterminedbyconservation ofcoffeesolids.)
c.Parameterisingtheequations Theparameterisationoftheequationsdescribedaboveisrelativelyquick butnotoptimal.Abetter(butmoretimeconsuming)approachwouldbetocarryoutleastsquaresfittingusing, forinstance,theexcelsolverorastatisticalenvironment suchasR.Thisapproach,aswellasgeneratingamore accuratefit,alsoallowsamorecomplexfunctiontobe usedforinstancetakingintoaccountthebimodalsize distribution.Becauseofthewidespacingofthepoints theshorttimescaleisdifficulttoresolvesoweadoptasingularperturbationstyleapproachandassumethatinan unresolvableboundarylayer15 theextractionyieldjumps rapidlyupfromzerotosomefinitevalue.Thisresultsin theequation
x = x1(1 e t/τ )+ x2(1 et/ τ )(17)
where x1 isthemaximumextractionpossiblefromlarge grainswhichoccursoveratimescale τ and x2 isthe maximumextractionpossiblefromsmallgrainswhich occursovera(verysmall)timescale τ .Inpractice cannotbeestimatedfromthedatasoinsteadthefunction x = x1(1 e t/τ )+ x2 isfittedtoallthedatapoints exceptfor t =0.Theresultingparametersasobtained bytheLevenbergMarquardtalgorithm16 aregiveninTableIIIandthefitisshowninFig.8.Thisproducesa slightlybetterfitandtheremainingcalculations(solving for toptimal and m/Mw )canstillbecarriedoutanalytically.
d.Visualisingthedata Anexcelworksheet,ora platformespeciallydesignedformathematicalvisualisationssuchasgeogebra,canbeincorporatedintothefittingprocess.Eithercouldbeusedtoobservetheimpact ofchangingthefittedparametersonthequalityofthe agreementwiththedataandcouldalsobelinkedtoadditionalplotsdemonstratinghowtheotherquantitiesof interestvary.
TABLEIII.Parametersofthemathematicalmodelofcoffee brewingobtainedbyLevenbergMarquardtleastsquaresfitting.Theuncertaintyinthefinaldigitisgiveninbracketsat theend.
x1 26 9(2) τ 24 1(1)s
∗ w.lee@hud.ac.uk;AlsoatMACSI,UniversityofLimerick
1 K.Sinnott, TheArtandCraftofCoffee:AnEnthusiast’s GuidetoSelecting,Roasting,andBrewingExquisiteCoffee (QuarryBooks,2010)iSBN-13:978-1592535637.
FIG.8.Leastsquaresfittodataincorporatingaboundary layerattheorigin.
V.CONCLUSIONS
Coffeeisanexcellentsystemtointroducestudentsto allthestepsofmathematicalmodellingfromsettingup amodeltousingthatmodeltomakepredictionsand improvementsandfinallytorelatingfindingsbacktothe realworldandimprovingpeopleslivesonecupofcoffee atatime.Coffeeisnotjustaninterestingdemonstration butalsolinksuptoanactiveareaofongoingresearch. ImplementationoftheactivityontheNUMBASplatform allowsstudentstoworkthroughthecalculationattheir ownpace.
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