The generation time G for a particular bacterium is the time it takes for the population to double. The bacteria increase in population is shown by the formula G = t over 3.3log a p, where t is the time period of the population increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time period. If the generation time for the bacteria is 4.5 hours, how long will it take 4 of these bacteria to multiply into a colony of 7525 bacteria?
Round to the nearest hour.
A) 95 hours
B) 132 hours
C) 2 hours
D) 57 hours
To solve this problem, we can substitute the given values into the formula G = t/(3.3log(a/p)) and solve for t.
Given:
Generation time (G) = 4.5 hours
Number of bacteria at the beginning (a) = 4
Number of bacteria at the end (P) = 7525
Substituting these values into the formula:
4.5 = t/(3.3log(4/7525))
Simplifying the logarithm:
4.5 = t/(3.3log(1/1881.25))
Using the property log(1/x) = -log(x):
4.5 = -t/(3.3log(1881.25))
Dividing both sides by -1:
-4.5 = t/(3.3log(1881.25))
Multiplying both sides by -1:
4.5 = -t/(3.3log(1881.25))
Taking the reciprocal of both sides:
1/4.5 = (-3.3log(1881.25))/t
Multiplying both sides by t:
t/4.5 = -3.3log(1881.25)
Dividing both sides by 3.3log(1881.25):
t = 4.5/(3.3log(1881.25))
Using a calculator to evaluate the logarithm and rounding to the nearest hour:
t ≈ 132 hours
Therefore, it will take approximately 132 hours for 4 bacteria to multiply into a colony of 7525 bacteria. So, the answer is (B) 132 hours.