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Math Help - A couple more optimization problems

  1. #1
    Junior Member NAPA55's Avatar
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    A couple more optimization problems

    Couldn't figure this one out for the life of me... got a few steps down and then it all went downhill from there.

    If the concentration of a drug in the bloodstream at time "t" is given by this function,
    where a, b(b>a), and k are positive constants that depend on the drug, at what time is the concentration at its highest level?

    And the second question:

    The motion of a particle is given by s(t) = 5cos(2t + pi/4). What are the maximum values of the displacement, the velocity, and the acceleration?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    I'd help

    but I cant read the paper...rewrite it...preferably in LaTeX
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by NAPA55 View Post
    Couldn't figure this one out for the life of me... got a few steps down and then it all went downhill from there.

    If the concentration of a drug in the bloodstream at time "t" is given by this function,
    where a, b(b>a), and k are positive constants that depend on the drug, at what time is the concentration at its highest level?

    And the second question:

    The motion of a particle is given by s(t) = 5cos(2t + pi/4). What are the maximum values of the displacement, the velocity, and the acceleration?

     c(t)=\frac{k}{b-a}\left[ e^{-at}-e^{-bt}\right]

    Taking the derivative we get...

     \frac{dc}{dt}=\frac{k}{b-a}\left[ -ae^{-at}+be^{-bt}\right]

    setting equal to zero and solving we get

    -ae^{-at}+ be^{-bt}=0 \iff \frac{a}{b}=\frac{e^{-bt}}{e^{-at}}\iff \frac{a}{b}=e^{(a-b)t} \iff ln \left( \frac{a}{b}\right)=(a-b)t

     t= \frac{ln \left( \frac{a}{b}\right)}{a-b}
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