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Math Help - Decay rates in exponential decay

  1. #1
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    Decay rates in exponential decay

    Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

    What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

    Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

    I'm thinking that could possibly boil down to

    0.01 = Alpha ( x ^ 1000)

    But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

    Any pointers?
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  2. #2
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    Quote Originally Posted by misterprogrammer View Post
    Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

    What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

    Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

    I'm thinking that could possibly boil down to

    0.01 = Alpha ( x ^ 1000)

    But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

    Any pointers?
    the way you have set up your equation, \alpha does not become 0.01, rather the product of \alpha and x^{1000} becomes 0.01

    it sounds like you want \alpha to be a given fixed constant 0.01 \le \alpha \le 1

    to solve for x ...

    0.01 = \alpha \cdot x^{1000}

    \frac{0.01}{\alpha} = x^{1000}

    \left(\frac{0.01}{\alpha}\right)^{\frac{1}{1000}} = x

    so ... say the given \alpha = 0.5

    \left(\frac{0.01}{0.5}\right)^{\frac{1}{1000}} = x

    (0.02)^{\frac{1}{1000}} = x

    x = 0.996095619...
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  3. #3
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    Hello, misterprogrammer!

    Your description is off.
    I hope I've interpreted your intentions correctly.


    I'm trying to write a computer program which invloves a half-life type of decay.

    What I want to calculate is a particular value, x to repeatedly multiply my initial amount A_o
    so that over 1000 steps A_o is reduced by a factor of 0.01

    Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.
    . . This is correct!

    We have: . A \:=\:A_ox^t

    When t = 1000, we want A to be 0.01 of A_o

    So, we have: . 0.01A_o \:=\:A_ox^{1000} \quad\Rightarrow\quad x^{1000} \:=\:0.01

    Therefore: . x \;=\;(0.01)^{\frac{1}{1000}} \;=\;0.995405417

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