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Math115/Exam1(February8,2022)page87.[7points]a.[4points]Zoey,azoologist,isstudyingthepopulationofgiraffesnearalake.Shenoticesthatthenumberofgiraffesnearthelakefluctuatesinasinusoidalmannerovera24hourcycle.Thegiraffepopulationreachesaminimumof30giraffesat7:00ameveryday,andrisestoamaximumof50giraffesat7:00pmeveryday.LetG(t)beasinusoidalfunctionmodelingthenumberofgiraffesatthelakethoursafter6:00am.FindaformulaforG(t).Solution:Becausewearegiventhetimesformaximumandminimumvalues,wechoosetousetransformationsofthecosinefunctiontomodelthissituation(thoughusingthesinefunctionrequiresonlyadifferenthorizontalshift).ThenG(t)hastheformG(t)=Acos(B(t−C))+DforsomeconstantsA,B,C,andD.Theamplitudeis(50−30)/2=10so|A|=10.Theperiodis24soB=π/12.Themidlineisaty=(50+30)/2=40soD=40.Finally,aminimumoccursatt=1sowecanuseaverticalreflection(giving−cos)combinedwithahorizontalshifttotherightby1.ThiswouldgiveC=1andA=−10.Therefore,onepossibleequationisG(t)=−10cos(π12(t−1))+40.Answer:G(t)=−10cosπ12(t−1)+40b.[3points]Zoeyalso...