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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

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(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]