BC Calculus 6.4 Practice worksheet

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