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Thread: Help with setting up a problem

  1. #1
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    Help with setting up a problem

    Hi, I need some help setting up this problem properly

    Problem:
    After 23 days a 10-milligram sample of radioactive material decays to 5 milligrams. After how many days will there be 1 milligram of the material?


    thanks!
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  2. #2
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    Quote Originally Posted by dibs View Post
    Hi, I need some help setting up this problem properly

    Problem:
    After 23 days a 10-milligram sample of radioactive material decays to 5 milligrams. After how many days will there be 1 milligram of the material?


    thanks!
    $\displaystyle m(t) = m_0 e^{- \lambda t}$ where m(t) is the mass of the material at a given time t, $\displaystyle m_0$ is the initial amount of mass m(0), $\displaystyle \lambda$ is the half-life, and t is the time elapsed.

    So we know that
    $\displaystyle 5 = 10 e^{- \lambda \cdot 23}$

    We need to solve this for $\displaystyle \lambda$.

    $\displaystyle \frac{1}{2} = e^{-23 \lambda}$

    $\displaystyle ln (1/2) = -23 \lambda$

    $\displaystyle \lambda = -\frac{ln(1/2)}{23} = \frac{ln(2)}{23} \approx 0.030136834$ /day

    So
    $\displaystyle m(t) = 10 e^{-0.030136834 t}$

    We wish to find out when m(t) = 1 mg.

    $\displaystyle 1 = 10 e^{-0.030136834 t}$

    Solve this like the preceding problem. I get $\displaystyle t \approx 76.404346182$ days.

    So 76.4 days or so.

    -Dan
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  3. #3
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    thank you very much for the help!
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