# Math Help - Two bacterial colonies

1. 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?

2. I have no idea because I don't know what K is defining. >

3. You need to think! Plug in t = 0 into the equation in post # 16. What do you get?

4. 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.

5. 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?

6. 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.

7. 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.

8. 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?

9. Hmm, this has stumped me, however I'm pretty sure it might have something to do with natural log...

10. 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!

11. Hmm, I have no idea how to find the other equation. Sorry for being stupid.

12. 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?

13. 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.

14. So, if P(2) = 2000, what does P(4) equal?

15. P(4) = 4000

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