The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies

Question

The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5549 years. Let C(t) be the amount of carbon-14 present at time t.
(a) Find the value of the constant k in the differential equation C’ = -kC.
(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material. How old was the Shroud of Turin at the time of this data?

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Thu Hương 5 years 2021-07-25T02:24:50+00:00 1 Answers 25 views 0

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  1. dangkhoa
    0
    2021-07-25T02:26:36+00:00
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    Answer:

    a) k = 0.00012491389

    b) The Shroud of Turin was 755 years old at the time of this data.

    Step-by-step explanation:

    (a) Find the value of the constant k in the differential equation C’ = -kC.

    First we find the differential equation, by separation of variables. So

    \int \frac{C^{\prime}}{C} dt = -\int k dt

    So

    \ln{C} = -kt + K

    In which K is the constant of integration, representing the initial amount of substance. So

    C(t) = C(0)e^{-kt}

    Half-life of 5549 years.

    This means that C(5549) = 0.5C(0). We use this to find k. So

    C(t) = C(0)e^{-kt}

    0.5C(0) = C(0)e^{-5549k}

    e^{-5549k} = 0.5

    \ln{e^{-5549k}} = \ln{0.5}

    -5549k = \ln{0.5}

    k = -\frac{\ln{0.5}}{5549}

    k = 0.00012491389

    So

    C(t) = C(0)e^{-0.00012491389t}

    (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material. How old was the Shroud of Turin at the time of this data?

    This is t for which C(t) = 0.91C(0)

    So

    C(t) = C(0)e^{-0.00012491389t}

    0.91C(0) = C(0)e^{-0.00012491389t}

    e^{-0.00012491389t} = 0.91

    \ln{e^{-0.00012491389t}} = \ln{0.91}

    -0.00012491389t = \ln{0.91}

    t = -\frac{\ln{0.91}}{0.00012491389}

    t = 755

    The Shroud of Turin was 755 years old at the time of this data.

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