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 , 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) 96 hours
B) 132 hours
C) 2 hours
D) 57 hours
To solve this problem, we first need to determine how many generations it will take for 4 bacteria to multiply into 7525 bacteria.
Let's start by finding the number of generations needed:
7525 = 4(2^(n))
7525/4 = 2^(n)
1881.25 = 2^(n)
Next, let's solve for n:
log2(1881.25) = n
n ≈ 11.23
Since we can't have a fraction of a generation, we will round up to 12 generations.
Since the generation time is 4.5 hours, the total time it will take for 4 bacteria to multiply into 7525 bacteria is:
12 generations * 4.5 hours/generation = 54 hours
Rounded to the nearest hour, it will take 54 hours for 4 bacteria to multiply into a colony of 7525 bacteria.
The closest answer choice is:
D) 57 hours