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. (1 point) Responses 95 hours 95 hours 132 hours 132 hours 2 hours 2 hours 57 hours
To solve this problem, we can use the formula for the population increase:
P = a * (2)^(t/G)
We are given that the generation time (G) is 4.5 hours. We want to find how long it will take for 4 bacteria (a) to multiply into a colony of 7525 bacteria (P). Let's substitute these values into the formula:
7525 = 4 * (2)^(t/4.5)
Dividing both sides by 4 gives:
1881.25 = (2)^(t/4.5)
To solve for t, we will take the logarithm (base 2) of both sides:
log2(1881.25) = t/4.5
Multiplying both sides by 4.5 gives:
t = 4.5 * log2(1881.25)
Using a calculator, we find that log2(1881.25) is approximately 10.915. Plugging this value into the equation gives:
t = 4.5 * 10.915 ≈ 49.118
Rounding to the nearest hour, it will take approximately 49 hours for 4 bacteria to multiply into a colony of 7525 bacteria. Therefore, the correct answer is 49 hours.