BC Calculus 6.4 Practice worksheet Name___________________________ 1. Oilisbeingpumpedcontinuouslyfromacertainoilwellatarateproportionaltothe dy amountofoilleftinthewell;thatis ky ,whereyistheamountofoilleftinthewell dt atanytimet.Initiallytherewere1,000,000gallonsofoilinthewell,and6yearslater therewere500,000gallonsremaining.Itwillnolongerbeprofitabletopumpoilwhen therearefewerthan50,000gallonsremaining. a) Writeanequationfory,theamountofoilremaininginthewellatanytimet. b) Atwhatrateistheamountofoilinthewelldecreasingwhenthereare600,000 gallonsofoilremaining? c) Inordernottolosemoney,atwhattimetshouldoilnolongerbepumpedfromthe well? 2. Useseparationofvariablestosolvetheinitialvalueproblem. dy cos x y 5 dx 3 y 2 BC Calculus 6.4 Practice worksheet Name___________________________ 3. LetP(t)representthenumberofwolvesinapopulationattimetyears,when t 0 . ThepopulationP(t)isincreasingataratedirectlyproportionalto800–P(t),wherethe constantofproportionalityisk. a) IfP(0)=500,findP(t)intermsoftandk. b) IfP(2)=700,findk. c) Find lim P (t ) . t 4.Ayamisputina2000Covenandheatsupaccordingtothedifferentialequation withkasapositiveconstant,tistime,inminutes,andHis temperaturein . (a)Iftheyamisat whenitisputintotheoven,solvethedifferentialequation. (b)Findkusingthefactthatafter30minutesthetemperatureoftheyamis . BC Calculus 6.4 Practice worksheet Name___________________________ 5.Adetectivefindsamurdervictimat9a.m.Thetemperatureofthebodyismeasuredat .Oneourlater,thetemperatureofthebodyis89 Thetemperatureoftheroom hasbeenmaintainedataconstant . (A)Assumingthatthetemperature,T,ofthebodyobeysNewton’sLawofCooling,writea dT k (T TS ) differentialequationforT. dt (B)Solvethedifferentialequationtoestimatethetimethemurderoccurred. 6.A66‐kgcyclistona7‐kgbicyclestartscoastingonlevelgroundat9m/sec.Thevalueof kis3.9kg/sec. V V0e k t m a)Abouthowfarwiththecyclistcoastbeforereachingacompletestop? b)Howlongwillittakethecyclist’sspeedtodropto1m/sec? BC Calculus 6.4 Practice worksheet Name___________________________ Multiplechoice 1.Thenumberofbacteriainacultureisgrowingatarateof perunitoftime,t. Att=0,thennumberofbacteriapresentwas2,000.Findthenumberpresentatt=4. (A) (B) (D) (E) (C) 2.ThechangeinN,thenumberofbacteriainaculturedishattimet,isgivenby . IfN=3whent=0,theapproximatevalueoftwhenN=1210is (A)2(B)3(C)4(D)5(E)6 3.Supposethat,duringthefirstyearafteritshatching,theweightofaduckincreasesata rateproportionaltoitsweight.Theducklingweighed2poundswhenitwashatchedand 3.5poundsatage4months.Howmanypoundswillthebirdweightatage6months? (A)4.2lbs(B)4.6lbs(C)4.8lbs(D)5.6lbs(E)6.5lbs 4.Therateofchangeofthesurfaceareaofacube,S,withrespecttotime,t,isdirectly proportionaltothesquarerootofone‐sixthofthesurfacearea.Whichofthefollowingisa differentialequationthatbestdescribestherelationship? (A) (D) (B) (E) (C) 5.Atanytime, ,indays,therateofgrowthofabacteriapopulationisgivenby ,where isthenumberofbacteriapresentand isaconstant.Theinitial populationis1,500andthepopulationquadrupledduringthefirst2days.Bywhatfactor willthepopulationhaveincreasedduringthefirst3days? (A)4(B)5(C)6(D)8(E)10
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