The​ half-life of polonium is 139​ days, but your sample will not be useful to you after 89​% of the radioactive nuclei present on the day t

Question

The​ half-life of polonium is 139​ days, but your sample will not be useful to you after 89​% of the radioactive nuclei present on the day the sample arrives has disintegrated. For about how many days after the sample arrives will you be able to use the​ polonium?

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Raelynn 4 weeks 2021-09-21T13:24:54+00:00 1 Answer 0

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    2021-09-21T13:26:04+00:00

    Answer:

    You will be able to use the sample for about 441 days.

    Step-by-step explanation:

    The equation for the amount of polonium after t days is given by:

    P(t) = P(0)e^{-rt}

    In which P(0) is the initial amount and r is the rate of decrease.

    The​ half-life of polonium is 139​ days

    This means that P(139) = 0.5P(0).

    We apply this to the equation, and find r.

    P(t) = P(0)e^{-rt}

    0.5P(0) = P(0)e^{-139r}

    e^{-139r} = 0.5

    Applying ln to both sides of the equality:

    \ln{e^{-139r}} = \ln{0.5}

    So

    -139r = \ln{0.5}

    139r = -\ln{0.5}

    r = -\frac{\ln{0.5}}{139}

    r = 0.005

    So

    P(t) = P(0)e^{-0.005t}

    Your sample will not be useful to you after 89​% of the radioactive nuclei present on the day the sample arrives has disintegrated. For about how many days after the sample arrives will you be able to use the​ polonium?

    It will be useful until t in which P(t) = 1-0.89 = 0.11P(0). So

    P(t) = P(0)e^{-0.005t}

    0.11P(0) = P(0)e^{-0.005t}

    e^{-0.005t} = 0.11

    Applying ln to both sides

    \ln{e^{-0.005t}} = \ln{0.11}

    -0.005t = \ln{0.11}

    0.005t = -\ln{0.11}

    t = -\frac{\ln{0.11}}{0.005}

    t = 441

    You will be able to use the sample for about 441 days.

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