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Question 387793: A biologist finds that the population of a certain type of bacteria doubles each half-hour. An initial culture has 90 bacteria.
How long will it take for the number of bacteria to reach 368,640?
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! first, let us make a table to see how the data behaves:
1/2 hour periods 0 1 2 3 4 .......
number of bacteria 90 180 360 720 1440
As stated in the problem, the above shows that the initial number of bacteria doubles each 1/2 hour. It can also be seen that we can calculate the number of bacteria reached at any period by multiplying the original amount of 90 by 2^period number. For example, the 720 reached after 3 periods is determined by multiplying the original amount of 90 by 2^3, after 4 periods by 2^4, after 2 periods by 2^2, etc.
If we call the original amount, P, and the amount reached after a certain number of 1/2 hour periods, A, and period number, n, we can come up with the following relationship:
A = P (2^n)
For this problem:
368640 = 90(2^n)
2^n=368640/90=4096
use logarithms to solve
n (log 2)=log 4096
n=log 4096/(log 2) = 12
ans: after 12 (1/2 hour) periods, or 6 hours, the bacteria count would have reached 368640
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