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Math Help - Half life problem

  1. #1
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    Half life problem

    Here is the question:
    An unknown radioactive element decays into non-radioactive substances. In days the radioactivity of a sample decreases by percent.

    (a) What is the half-life of the element?
    (b)How long will it take for a sample of mg to decay to mg?

    I've tried it several times and I keep getting the same answers which are wrong

    I used dy/dx=-ky I integrate this and got y=ce^{-kt}.

    I used y(0)=y_0 which gave me the equation  y=y_0e^{-kt}.

    I then plugged in 0.37y_0 for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation y(t)=y_0e^{-.0025t}.

    I then set y(t)= 0.5y_0 and solved for t to get the half life. Which gave me the answer 277 days.

    What am I doing wrong?
    Last edited by mr fantastic; September 4th 2009 at 03:09 PM. Reason: Fixed latex and improved formatting for easier readability.
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  2. #2
    MHF Contributor
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    1) Why do you think it is wrong?

    2) Check your notation. How do you get from dy/dx = -ky to y = f(t)? It's okay, since everyone knows what you mean, but you should write it as clearly as you can.

    3) You rounded 'k' rather severely. Use more decimal places.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by cmerickson21 View Post
    Here is the question:
    An unknown radioactive element decays into non-radioactive substances. In days the radioactivity of a sample decreases by percent.

    (a) What is the half-life of the element?
    (b)How long will it take for a sample of mg to decay to mg?

    I've tried it several times and I keep getting the same answers which are wrong

    I used dy/dx=-ky I integrate this and got y=ce^{-kt}.

    I used y(0)=y_0 which gave me the equation  y=y_0e^{-kt}.

    I then plugged in 0.37y_0 for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation y(t)=y_0e^{-.0025t}.

    I then set y(t)= 0.5y_0 and solved for t to get the half life. Which gave me the answer 277 days.

    What am I doing wrong?
    I always find this approach to half life problems bizarrely over complex.

    If half-life means anything it tells you that if you start at t=0 with an initial quantity m_0 which decays (exponentially) with half-life t_{1/2}, then the quantity remaining at t is:

    m=m_0 2^{-\frac{t}{t_{1/2}}}

    CB
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