"A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute, and with the amount she currently has, she calculates that the bottle will be full in one hour. She sterilizes the bottle and places the culture in the bottle at 11 am.
a) At What time will the bottle be half full?"
2 = e^k, k = ln(2)
1/2 = e^(t*ln(2)), t = ln(1/2)/ln(2) = -1
I thought I understood the question and did the work pretty easily, but I don't understand the answer I'm getting. How does -1 minutes make sense here?
Best Answer
This is a question that shows the bad guessing abilities of the human when it comes to exponential growth. No calculation is required here - the bottle will be half full after 59 minutes, since it doubles every minute it will be full at minute 60 which equals 1 hour. So the bottle will be half full at 11.59 am.
As an alternative you can also calculate it: Since you double every minute, you can choose $y$ as the amount of "bacteria units" in your bottle and $t$ as the amount of minutes gone. That gives you the equation $y = 2^t$ , after one hour (60 minutes) you then have $2^{60}$ of "bacteria units" , to solve when you get half of it you then take $\frac{2^{60}}{2}$ which is $2^{59}$ , so 59 minutes.