Question

This exercise uses the population growth model. The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours. What is the relative rate of growth of the bacteria population? Express your answer as a percentage.

Answer 1

To find the relative rate of growth of a bacteria population that grew from 400 to 25,600 bacteria in 4 hours, use the exponential growth formula and calculate the growth rate 'r', then express it as a percentage.

Step-by-step explanation:

The student is working with a population growth model to determine the relative rate of growth of a bacteria population. Given the population of bacteria increases from 400 to 25,600 over 4 hours (from the 2nd to the 6th hour), we can use the exponential growth formula P(t) = P0ert, where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm, and r is the relative rate of growth. We can calculate r by taking the natural logarithm of the ratio of the final population to the initial population, divided by the time interval. In this case:

1. Determine the time interval, which is 6 hours - 2 hours = 4 hours.
2. Use the formula ln(P(t)/P0) / t = r, where P(t) is 25,600, P0 is 400, and t is 4 hours.
3. Calculate r and then multiply by 100 to get the percentage rate.

Upon calculation, you will find the relative rate of growth of the bacteria population. This process is based on understanding exponential growth, which characterizes populations like bacteria that have the capacity to double in number at regular intervals.