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Math Help - Population Models

  1. #1
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    Population Models

    Hey,

    I'm having trouble with a population model question. I have been given the differential equation dN/dt=kN; where t is the time elapsed in hours and k is a constant.

    I have also been given some initial conditions; when t=0, N=3000 and when t=3, N=9000

    With these values i have been asked to find the value of K

    I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3), does this look right?

    Also i have been given the following:
    For t > 3 the culture is washed with a solution that is harmful to the bacteria, and the bacteria are killed of 4500 per hour.

    Using this i have been asked to find the number of bacteria in the culture for any time where t>3

    I am completely stuck for this part of the question!

    Any ideas??

    Thanks, Function.
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  2. #2
    tdw
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    Dude I have the same problem.

    I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3) too

    but the second part is confusing.
    I tried to solve again by dN/dt= kN - 4500

    but failed to solve......
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  3. #3
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    Hi function boy,

    Did you say you had a problem with changing gears? I might know someone who can help you out


    Thanks, Function.
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  4. #4
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    Quote Originally Posted by tdw View Post
    Dude I have the same problem.

    I found the equation to be B(t)=3000e^kt, where K= (1/3)ln(3) too

    but the second part is confusing.
    I tried to solve again by dN/dt= kN - 4500

    but failed to solve......
    \frac{dN}{kN- 4500}= dt
    Integrating both sides, \frac{1}{k}ln(kN- 4500)= t+ C so ln(kN- 4500)= kt+ kC and then kN- 4500= e^{kC}e^{kt}= C'e^{kt} where C'= e^{kC}.
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  5. #5
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    Thanks,

    I follow what you have done there, but how do i go about solving the variables K and C???? I have tried to use the initial conditions but i dont think that is right, becuase this equation is only for t > 3.

    Function
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  6. #6
    tdw
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    Hey function,
    I think the intial condition would be t=3 which is N=9000
    coz time starts from t=3 so from t=3 on t= 0
    haha.... I dunno whether am I right, that the only way I can think about.
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  7. #7
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    Hey tdw,
    That's what i thought aswell, but that will only give you one variable, and when i tried to do it it just turned into a huge mess . Did you get an answer in the end?
    Function
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