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MTH138Exponential/LogModels:Radioactivity
BeginningwithChapter4,weintroducedexponentialandlogarithmicfunctions,discoveredsomeoftheir
properties,anddevelopedtechniquesforsolvinglogarithmicandexponentialequations.InSection4.5,we
discussedsomeapplicationsandmodelsinvolvingthesenewfunctions.Oneparticularmodelisusedtostudythe
rateatwhichradioactivematerialdecaysintomorestableforms.
I.
RadioactiveDecayFormula
• Radioactivedecaycanbemodeledusingavariationofthegeneralexponentialdecaymodel:
𝐴 = 𝐴! 𝑒 !" ,where
• 𝐴! istheinitialamountofmaterialpresent
• 𝑘isthedecayconstant(uniqueforeachelement)
• 𝑡istime(inyears,days,minutes,etc)
• 𝐴istheamountofmaterialremainingaftersometimehaspassed
• Rememberthat𝑒(Euler’sNumber)isnotavariable.
II.
WhatisHalf-Life?
Thehalf-lifeofanelementreferstotheamountoftimeitwouldtakeforhalfoftheinitialamounttodecay.For
example,supposeyouwereholding1gramofRadon-219,aradioactiveisotopeoftheelementradon.Inlesstime
thanittakesyoutoreadthissentence,halfofyourradonwouldhavedecayedintoanotherelement(thehalflife
ofRadon-219is3.96seconds).Someelementshavehalf-lifesmeasuredinmillionthsofasecond,whileothersare
measuredinbillionsofyears.
Foravisualrepresentationofdecayandhalf-life,trytheappletonthefollowingwebsite:
http://www.colorado.edu/physics/2000/isotopes/radioactive_decay3.html
III.
Examples
A. FindthedecayrateofPlutonium-241,whichhasahalf-lifeof14.4years.
B. If24 𝑔ofPlutonium-241werepresentinitially,howlongwouldittakeforallbut3 𝑔todecay?
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C.
HowmuchPlutonium-241waspresentinitiallyif9.2 𝑘𝑔remainsafter30years?
IV.
Nowyoudoit!
A. Cobalt-57hasadecayconstantof𝑘 ≈ −.00256.If46 𝑘𝑔arepresentinitially,howmuchremainsafter92
days?Note:thetimeunitwillbedays
B. Cobalt-57hasadecayconstantof𝑘 ≈ −.00256.Findthehalf-lifeofCobalt-57.Note:thetimeunitwillbe
days
C. Fromaninitialquantityof36.6 𝑔ofCalifornium-252,12.2 𝑔remainsafter4.18years.Howmuchremains
afteranadditional7years?
D. ArchaeologistscanusetheknowndecayrateofCarbon-14toestimatetheageoforganicobjects.Suppose
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IllinoisJones hasdiscoveredawoodenspearinacave.Ifthewoodcontains12%ofitsoriginalamountof
Carbon-14,approximatelyhowoldisthespear?Thehalf-lifeofCarbon-14is5730years
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