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Math Help - Exponential Decay

  1. #1
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    Exponential Decay

    A first order exponential decay can be written as
    A(t)= Ae^(-t/r)
    where A(t) is the amount (of substance) after time t, A is the initial amount at time t=0 and r is the decay time.

    The fall time is defined as the time in which A(t) falls from 90% to 10% of its initial value. Find the relationship between the fall time of an exponential decay and r.
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  2. #2
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    .9A = Ae^{-\frac{t_1}{r}}

    .9 = e^{-\frac{t_1}{r}}

    \ln(.9) = -\frac{t_1}{r}

    t_1 = -\frac{\ln(.9)}{r}

    using the same algebra ...

    t_2 = -\frac{\ln(.1)}{r}

    \Delta t = t_2 - t_1 = -\frac{\ln(.1)}{r} + \frac{\ln(.9)}{r}

    \Delta t = \frac{\ln(9)}{r}
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