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Math Help - An application in growth

  1. #1
    No one in Particular VonNemo19's Avatar
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    An application in growth

    The population of a bacterial culture grows at a rate proportional to the number of bacteria present at any time. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 present. What was the initial number of bacteria?

    My attempt:

    It is implied by context that a model for the rate of growth is

    \frac{dP}{dt}=kP which implies that P(t)=ce^{kt}.

    But the fact that P(3)=200, and P(10)=2000 we see that

    400=ce^{3k} and 2000=ce^{10k}

    Solving for c in the first equation gives

    c=400e^{-3k}. Substituting into the second...

    2000=400e^{-3k}e^{10k} and we see that k=\frac{\ln5}{7}

    Now, again using the fact that P(3)=400 we can conclude that c=400e^{(-3\ln5)/7}

    So that P(t)=400e^{\frac{\ln5}{7}(t-3)}

    and P(0)=400e^{\frac{\ln5}{7}(0-3)}

    Have I done this right. If so, was there a faster way?
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  2. #2
    A Plied Mathematician
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    Everything looks good to me. The only thing that might have speeded things up a hair, would have been if you had divided one equation by the other. That would have gotten rid of the c a little quicker. There's not a lot of fluff in your solution, though.
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