1. ## Two bacterial colonies

Two bacteria colonies are cultivated in a laboratory. The first colony has a doubling time of 2 hours and the second a doubling time of 3 hours. Initially, the first colony contains 1000 bacteria and the second colony 3,000 bacteria. At what time t will sizes of the colonies be equal?

We never learned this in class, and he expects us to know it for the next one. I have no idea what to do here. :S

2. What kind of function doubles when you increase t by a certain amount each time?

3. By researching a bit, I have come to the conclusion that this is an exponential growth function. We never learned this, lol.

4. You are correct. Can you figure out exactly which functions will fit your conditions? (You'll need two separate functions here, one for each colony.)

5. Hmm, nope, I have no clue.

6. Ok. Well, let's just take the first colony. You know that its initial population is 1000, and it has a doubling time of 2 hours. I would posit a function of the form

$\displaystyle P(t)=K\,n^{kt}.$

What happens when you plug in t = 0? What does that say about the value of K?

7. It basically says that when t = 0, the answer will be K. I'm guessing k has a direct relationship with P

8. Well, yes, it does have a direct relationship. When t = 0, your function returns K. But you know the population when t = 0, don't you?

9. The population would be 0 would it not? P(0) = 0.

10. No, no. This is the population when you start the clock. The initial population. What is that?

11. 1,000 or 3,000, depending on the colony.

12. Right. Well, we were looking at just the first colony, as per Post # 6. So, what does this tell you about the equation in Post # 6 for the first colony?

13. When t = 0, the population will be 1,000.

14. So K is... what?

15. K is the new population after a certain amount of time.

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