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Math Help - Two bacterial colonies

  1. #16
    A Plied Mathematician
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    No, no. The equation for P = P(t) as a population that depends on time is

    P(t)=K\,n^{rt}.

    I've changed the small k to an r to avoid confusion. K, n, and r are constants. So, what is K for the first population?
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  2. #17
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    I have no idea because I don't know what K is defining. >
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  3. #18
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    You need to think! Plug in t = 0 into the equation in post # 16. What do you get?
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  4. #19
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    You get K which is equal to 1,000. That's all I can come up with unless I'm missing something. Boy do I feel stupid right now.
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  5. #20
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    You shouldn't feel stupid! K = 1000 is exactly correct. That tells you that the interpretation of K is that it is the initial population. So, perhaps a more telling modification of the original equation would be this:

    P(t)=P_{0}\,n^{rt}.

    Here P_{0}=P(0), the initial population.

    Ok. We've narrowed down our function by one constant. We now need to get a handle on n and r. You need to choose them such that P(2)=2000, for the first colony. Also, you'd need P(4)=4000, P(6)=8000, etc. How do you suppose you can do that?
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  6. #21
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    Just out of curiosity, do you know about geometric progression and the formula for the nth term?

    This might be an easier way for you to solve this problem, although I would advise that you try the method Ackbeet put forward because it helps you to think and construct your own equations.
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  7. #22
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    Well the equation would be P(2) = 1000n^2^r I'm guessing, and you would have to solve for the variables? This problem is confusing me.
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  8. #23
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    You could actually write

    2000=1000\,n^{2r}, couldn't you? You know the population when t=2: it's doubled from the initial population.

    However, that's one equation with two unknowns. You'd like one more equation. How can you get it?
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  9. #24
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    Hmm, this has stumped me, however I'm pretty sure it might have something to do with natural log...
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  10. #25
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    Well, you will certainly need the natural logarithm. However, what I'm asking you is how can you get another equation? One equation is not enough to solve for two variables!
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  11. #26
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    Hmm, I have no idea how to find the other equation. Sorry for being stupid.
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  12. #27
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    You're being too hard on yourself. I've actually already given you, basically, the information you need. Take another look at post # 20. Can you see how to get another equation from that post?
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  13. #28
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    From post #20, I understand that when the time increases by 2, the population doubles. I'm just having a hard time changing it into an equation.
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  14. #29
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    So, if P(2) = 2000, what does P(4) equal?
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  15. #30
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    P(4) = 4000
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