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Prove some properties of a substance that decays at a rate proportional to the amount present

Consider a substance which disintegrates at a rate proportional to the amount present. Let y = f(t) denote the amount of the substance present at time t. Define the 1/nth life of the substance for a positive integer n as the number T such that

    \[ f(T) = \frac{f(0)}{n}. \]

  1. Prove that the 1/nth life is the same for any given sample of the substance (i.e., prove that the 1/nth life is invariant under the starting size of the sample), and compute T in terms of n and the decay constant k.
  2. For given real numbers a and b, prove we can write the function f(t) as

        \[ f(t) = f(a)^{w(t)} f(b)^{1-w(t)} \]

    and determine the function w(t).


  1. Proof. Since

        \[ f(T) = \frac{f(0)}{n} \]

    we have

        \[ \frac{f(T)}{f(0)} = \frac{1}{n} \quad \implies \quad \frac{f(0) e^{-kT}}{f(0)} = \frac{1}{n} \quad \implies \quad e^{-kT} = \frac{1}{n}. \]

    This is independent of the initial sample (since it depends only on the 1/nth life T and the decay constant k). Further, computing the 1/nth life in terms of n and the decay constant we have

        \[ e^{-kT} = \frac{1}{n} \quad \implies \quad -kT = -\log n \quad \implies \quad T = \frac{\log n}{k}. \qquad \blacksquare\]

  2. Proof. Let

        \[ w(t) = \frac{b-t}{b-a}. \]

    Then,

        \begin{align*}  f(a)^{w(t)} f(b)^{1-w(t)} &= e^{-ka\left( \frac{b-t}{b-a} \right)} \cdot e^{-kb \left( 1 - \frac{b-t}{b-a} \right)} \\[9pt]  &= \exp \left(- \frac{kab-kat}{b-a} - kb + \frac{kb^2-kbt}{b-a}} \right) \\[9pt]  &= \exp \left( \left( \frac{1}{b-a} \right) \left( -kab +kat - kb^2 + kab + kb^2 -kbt \right) \right) \\[9pt]  &= \exp \left( \left( \frac{1}{b-a} \right) (kat - kbt) \right) \\[9pt]  &= e^{-kt} \\  & = f(t). \qquad \blacksquare \end{align*}

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