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Math Help - A problem of half lives

  1. #1
    Junior Member
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    A problem of half lives

    Hey guys, I'm stuck on a problem:

    The concentration C of a drug in the bloodstream decays exponentially according to the equation

    C(t) = (D/V)(e^-kt)

    where t is time, D is initial dosage in mg, V is the volume in litres and k is a rate constant.

    A drug is said to be active if the concentration in the bloodstream is above some minimum concentration C(min). The duration of response t_duration is the amount of time for which a drug is active in the bloodstream.

    I need to prove that the duration of response increases by one half life when the initial dosage is doubled.

    Anybody know how I would approach this problem?
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  2. #2
    MHF Contributor
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    Re: A problem of half lives

    If t_h is half-life, then \frac{D}{V}e^{-kt_h}=\frac{1}{2}\frac{D}{V}.

    If t_d is the duration of response for the initial concentration D / V and t is the increase of the duration of response for the initial concentration 2D / V, then \frac{2D}{V}e^{-k(t_d+t)}=C_{\text{min}}=\frac{D}{V}e^{-kt_d}. From here, compare t with t_h.
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  3. #3
    Junior Member
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    Re: A problem of half lives

    Thanks a lot! I was struggling on this problem for a long time
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