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CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visit
ibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.
Radioactivedecay[39marks]
1.[Maximummarks:31]
Wecanobtainadiscretemodelofradioactivedecaybycollectivelyrollinganumber
ofdiceandthenaftereachrollremovingalldiceshowingasix,beforerepeatingthe
processwiththediceleft.
(a) Wedefinethenumberofdiceleftafter!rollsby! ! . Ifwestartwith!!dice,findanequationfor! ! .
[2]
(b) Byconsideringtherelationship! = !!" (!),findanequationfor! ! inthe
form!!!!!"where!isaconstantyoushouldfind.[3]
Thecontinuousradioactivedecayofatomscanbemodeledwiththefollowing
equation:
! ! = !!!!!"
! ! :Thequantityoftheelementremainingaftertimet(years).!!: Theinitialquantityoftheelement.!: Theradioactivedecayconstant.
(c) Carbon-14hasahalf-lifeof5730years.Thismeansthatafter5730years
exactlyhalfoftheatomsoftheoriginalquantitywillhavedecayed.Usethis
informationtofindthe!, theradioactivedecayconstantofCarbon-14.[3]
(d) YoufindanoldmanuscriptandaftertestingthelevelsofCarbon-14youfind
thatitcontainsonly30%ofCarbon-14ofanewpieceofpaper.Howoldis
thispaper?
[2]
(e) Ifwedefine ! ! !" =! ! ,wecanevaluateimproperintegralsasfollows:
! ! !" = lim!→!
! ! !"!
!
!
!= lim
!→![! ! ]!!
Showthat!!! !" = 1!
!
[4]
CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visit
ibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.
(e) Theprobabilitydensityfunctionfortheprobabilityofradioactivedecayof
Carbon-14canbegivenby:
! ! = !"!!" , ! ≥ 0.
Byconsidering ! ! !"!! ,showthat! = !.
[6]
(f) Henceshowthatthemedianfortheprobabilitydensityfunctiondoesgive
5730yearsto3significantfigures.
[3]
(g) UsecalculustofindthemeanlengthoftimeaCarbon-14atomwillexistbeforedecaying.
[7]
(ii) Commentonyourresult.
[1]
2.[Maximummarks:8]
Inthisquestionweexploreradioactivedecaychains.Inadecaychain,atomAwill
decaytoatomB,whichthendecaystoatomCetc.Inourcasewewillsaythat
Ramanujan-1729decaysintoRamanujan-1728,whichthendecaysintoRamanujan-
1727.
(a) Westartwith100atomsofRamanujan-1729withadecayconstant!! =!
!"#$.Ramanujan-1728hasdecayconstant!! =!
!"#!.Thereforewehavethe
followingdifferentialequationfortherateofchangeofRamanujan-1728,!!:
!!!!" = − 1
4104!! +1
1729 100 !!!
!"#$!
Giventhatwhen! = 0,!! = 0 UseEuler’swithstepsize0.1tofindanapproximationfor!! when! = 0.5.
[6]
(b) Thesolutiontothedifferentialequationaboveisgivenby:
!! ! = 1001729
14104−
11729
!!!
!"#$! − !!!
!"#!!
Usetheequationabovetofind!! 0.5 andcommentontheaccuracyofyourapproximation.
[3]