# Two bacterial colonies

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• November 5th 2010, 09:54 AM
lancelot854
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
• November 5th 2010, 09:58 AM
Ackbeet
What kind of function doubles when you increase t by a certain amount each time?
• November 5th 2010, 10:00 AM
lancelot854
By researching a bit, I have come to the conclusion that this is an exponential growth function. We never learned this, lol.
• November 5th 2010, 10:05 AM
Ackbeet
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.)
• November 5th 2010, 10:08 AM
lancelot854
Hmm, nope, I have no clue. :(
• November 5th 2010, 10:11 AM
Ackbeet
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

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

What happens when you plug in t = 0? What does that say about the value of K?
• November 5th 2010, 10:19 AM
lancelot854
It basically says that when t = 0, the answer will be K. I'm guessing k has a direct relationship with P
• November 5th 2010, 10:20 AM
Ackbeet
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?
• November 5th 2010, 10:25 AM
lancelot854
The population would be 0 would it not? P(0) = 0.
• November 5th 2010, 10:26 AM
Ackbeet
No, no. This is the population when you start the clock. The initial population. What is that?
• November 5th 2010, 10:34 AM
lancelot854
1,000 or 3,000, depending on the colony.
• November 5th 2010, 10:35 AM
Ackbeet
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?
• November 5th 2010, 10:40 AM
lancelot854
When t = 0, the population will be 1,000.
• November 5th 2010, 10:41 AM
Ackbeet
So K is... what?
• November 5th 2010, 10:43 AM
lancelot854
K is the new population after a certain amount of time.
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