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Math Help - Population Question

  1. #1
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    Population Question

    A population, P, is said to be growing logistically if the time, T, taken for it to increase from P_1 and P_2 is given by:

    T=\int_{P_1}^{P_2} \frac {kdP}{P(L-P)}

    where k and L are positive constants and P_1 < P_2 < L.

    A) Calculate the time taken for the population to grow from P_1 = \frac {L}{4} to P_2 = \frac {L}{2}.

    B)What happens to T as P_2 approaches L?

    I honestly have no clue how to even start this problem. Any help would be appreciated.

    Thanks!
    Last edited by tiace; March 18th 2010 at 10:32 PM. Reason: Error in the integral
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  2. #2
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    Quote Originally Posted by tiace View Post
    A population, P, is said to be growing logistically if the time, T, taken for it to increase from P_1 and P_2 is given by:

    \int_{P_1}^{P_2} \frac {kdP}{P(L-P)}

    where k and L are positive constants and P_1 < P_2 < L.

    A) Calculate the time taken for the population to grow from P_1 = \frac {L}{4} to P_2 = \frac {L}{2}.

    B)What happens to T as P_2 approaches L?

    I honestly have no clue how to even start this problem. Any help would be appreciated.

    Thanks!
    To do the integration you'll need to use Partial Fractions. Then substitute P_1 = \frac{L}{4} and P_2 = \frac{L}{2} as your terminals.
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  3. #3
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    Is this integral correct?

    T=\int_{P_1}^{P_2} \frac {kdP}{P(L-P)} = \frac {-k}{L} * (\ln|P-L|-\ln|P|)

    I'm not sure where to go from here though.
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  4. #4
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    Yes that's fine.

    So now substitute your terminals.
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  5. #5
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    I'm not sure what you mean by substitute your terminals.

    Do you mean:

    T=(\frac {-k}{L} * (\ln|\frac{L}{2}-L|-\ln|\frac{L}{2}|))-(\frac {-k}{L} * (\ln|\frac{L}{4}-L|-\ln|\frac{L}{4}|))

    Which simplifies to:

    T=\frac{k\ln{3}}{L}
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