Final answer:
To find the time for a bacterial population to increase from 5,000 to 50,000 cells during its exponential growth phase, we first calculate the growth rate and then use it in the formula for exponential growth to solve for the required time.
Step-by-step explanation:
The question involves calculating the time it'll take for a bacterial population to grow to a certain quantity using the concept of exponential growth. Given the initial population of 5,000 bacteria grows to 12,000 bacteria in 10 hours, we can determine the growth factor per hour and use it to calculate the required time for the population to reach 50,000.
First, let's calculate the growth factor using the formula for exponential growth, which is P(t) = P0 × e^(rt), where:
- P(t) is the future value of the population
- P0 is the initial population
- r is the growth rate
- t is the time in hours
By inputting our known values into the formula, we get 12,000 = 5,000 × e^(10r), which can be rearranged to solve for r. Once we find r, we can then put it back into the formula along with the desired final population of 50,000 to solve for t, the time needed for the population to reach 50,000. As a matter of checking, we implicitly use the rule of 70 to understand how quickly the population doubles, though the rule isn't directly applicable to solving this problem.