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.

Question

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.

Answer 1

First, let's set up the formula for the population increase:

P = a * 2^(t/G)

We are given that the generation time G is 4.5 hours, the initial number of bacteria a is 4, and the final number of bacteria P is 7525.

Plugging in these values, we get:

7525 = 4 * 2^(t/4.5)

Divide by 4 on both sides:

1881.25 = 2^(t/4.5)

Take the logarithm base 2 of both sides to solve for t:

log2(1881.25) = t/4.5

t = 4.5 * log2(1881.25)
t ≈ 4.5 * 11 = 49.5

So, it will take approximately 49.5 hours for 4 bacteria to multiply into a colony of 7525 bacteria. Rounded to the nearest hour, it will take 50 hours.