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Math Help - Radio active decay

  1. #1
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    Radio active decay

    Radioactive substance has a half life of 18 years. Decay rate remains constant how long will it before 95 percent of the sample has decayed?

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  2. #2
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    Hello, av8or91!

    A radioactive substance has a half life of 18 years. Decay rate remains constant.
    How long will it be before 95 percent of the sample has decayed?
    The half-life function is: . A \;=\;A_o\,e^{-kt}
    . . where A_o is the intial amount and . k \:=\:\frac{\ln(2)}{\text{half-life}}

    Our function is: . A \;=\;A_o\,e^{-\frac{\ln 2}{18}t}


    If 95% of the sample has decayed, then: . A \,=\,0.05A_o

    We have: . A_oe^{-\frac{\ln2}{18}t} \:=\:0.05A_o \quad\Rightarrow\quad e^{-\frac{\ln2}{18}t} \:=\:0.05

    Take logs: . \ln\left(e^{-\frac{\ln2}{18}t}\right) \:=\:\ln(0.05) \quad\Rightarrow\quad -\tfrac{\ln2}{18}t\ln(e) \:=\:\ln(0.05)

    . . t \:=\:\frac{18\ln(0.06)}{-\ln2} \;=\; 77.79470571 \;\approx\;77.8\text{ years}

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