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Math Help - Using natural logoritums

  1. #1
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    Using natural logoritums

    Ok, here is my question

    In a laboratory experiment a quantity of a radioactive isotope was placed in a closed container. Measurements were taken to determine how much of the isotope remained after various intervals of time. The experimenter omitted to provide information on the initial mass M0 (0 is small at bottom of M) of the isotope, but postulated that M and M0 were related by the decay equation:

    M = M0e-kt (-kt is small and at the top, as if its to the power of -kt).

    where k is the decay constant and t is time. Show how to transform the equation by taking natural logarithms so that a straight line graph of the form y = mx + c may be plotted, and state how y , m and c are related to M , M0 , k and t .

    Not too sure how to go about this. Can someone please give me some guidence?
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  2. #2
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    Quote Originally Posted by oli212 View Post
    Ok, here is my question

    In a laboratory experiment a quantity of a radioactive isotope was placed in a closed container. Measurements were taken to determine how much of the isotope remained after various intervals of time. The experimenter omitted to provide information on the initial mass M0 (0 is small at bottom of M) of the isotope, but postulated that M and M0 were related by the decay equation:

    M = M0e-kt (-kt is small and at the top, as if its to the power of -kt).

    where k is the decay constant and t is time. Show how to transform the equation by taking natural logarithms so that a straight line graph of the form y = mx + c may be plotted, and state how y , m and c are related to M , M0 , k and t .

    Not too sure how to go about this. Can someone please give me some guidence?

    You have a decay equation:

    M(t)=M_0 e^{-kt}

    Now take logs:

    \ln(M)=\ln(M_0)-kt

    So put y=\ln(M), ...

    RonL
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  3. #3
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    Ok thanks,

    So just to confirm,

    Y would = ln (M)
    M would = ln(M0)
    and C would = t

    ????
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  4. #4
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    Quote Originally Posted by oli212 View Post
    Ok thanks,

    So just to confirm,

    Y would = ln (M)
    M would = ln(M0)
    and C would = t

    ????
    No y is \ln(M), x is t, m is -k, and c is \ln(M_0)

    RonL
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  5. #5
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    Ah i see.

    Is there any chance you could just quickly run through how you got there?

    Thanks in advance.
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  6. #6
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    Also, the question asks to estimate the value of the decay constant k.

    What would that be?
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by oli212 View Post
    Also, the question asks to estimate the value of the decay constant k.

    What would that be?

    You plot a curve of \ln(M) against t. The slope
    of the line of best fit is the estimate of -k

    RonL
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