1 Doubling Time De64257nition of doubling time The time required for each doubling in exponential growth is called the doubling time After a time an exponentially growing quantity with a doubling time double increases in size by a factor of double ID: 36777
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Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionB Ex.2Compoundinterestisaformofexponentialgrowthbecauseaninterestbearingaccountgrowsbythesamepercentageeachyear.Supposeyourbankaccounthasadoublingtimeof13years,bywhatfactordoesyourbalanceincreasein50years? Ex.3Worldpopulationgrowth.Worldpopulationdoubledfrom3billionin1960to6billionin2000.Supposethatworldpopulationcontinuestogrowwithadoublingtimeof40years.Whatwillthepopulationbein2030?in2200?2 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionB SectionB.3:ExponentialDecayandHalf-Life Exponentialdecayandhalf-life Exponentialdecayoccurswheneveraquantitydecreasesbythesamepercentageineveryxedtimeperiod.Inthatcase,thevalueofthequantityrepeatedlydecreasestohalfitsvalue,witheachhalvingoccurringinatimecalledthehalf-life=Thalf�life.Afteratimet,anexponentiallydecayingquantitywithahalf-timetimeThalf�lifedecreasesinsizebyafactorof1 2t Thalf�lifeThenewvalueofthedecayingquantityisrelatedtoitsinitialvalue(att=0)bynewvalue=initialvalue1 2t Thalf�life Ex.6Youmayhaveheardhalf-livesdescribedforradioactivematerialssuchasuraniumorplutonium.Forexample,radioactiveplutonium-239(Pu-239)hasahalf-lifeofabout24;000years.Supposethat100poundsofPu-239isdepositedatanuclearwastesite.Howmuchplutoniumwehaveafter24;000years?Andafter48;000years?Andafter72;000years?4 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionB SectionB.4:TheApproximateHalf-LifeFormula Denitionofapproximatehalf-lifeformula Theapproximatedoublingtimeformula(theruleof70)foundearlierworksequallywellforexponentialdecayifwereplacethedoublingtimewiththehalf-lifeandthepercentagegrowthratewiththepercentagedecayrate.ForaquantitydecayingexponentiallyatarateofP%pertimeperiod,thehalf-lifeisapproxi-matelyThalf�life70 PThisapproximationworksbestforsmalldecayratesandbreaksdownfordecayratesoverabout15%. Ex.9SupposethatinationcausesthevalueoftheRussianrubletofallatarateof12%peryear(relativetothedollar).Approximatelyhowlongdoesittakefortherubletolosehalfitsvalue?6 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionB Ex.11SupposetheRussianrubleisfallinginvalueagainstthedollarat12%peryear.Usingtheexacthalf-lifeformula,determinehowlongittakestherubletolosehalfitsvalue.8