Final answer:
The relative growth rate for the E. coli culture, with an initial population of 60 cells doubling every 20 minutes, is approximately 2.079 per hour. This is derived using the exponential growth formula and taking the natural logarithm of the ratio between the population after one hour and the initial population.
Step-by-step explanation:
The question asks to find the relative growth rate of a culture of Escherichia coli (E. coli) bacteria which doubles in number every 20 minutes. We assume the initial population is 60 cells and the time is measured in hours.
Firstly, we need to find out how many times the bacteria will divide in one hour. Since there are 60 minutes in an hour, the bacteria will divide 3 times (60 / 20 = 3). Being an exponential growth, the bacteria population doubles with each division.
The formula for exponential growth rate is given by N(t) = N0 * e^(rt), where N(t) is the population at time t, N0 is the initial population, e is the base of the natural logarithm, and r is the relative growth rate. In this situation, after one hour (t=1), our population would have doubled three times. This gives us 60 * 2^3 which equals 480.
Therefore, we can now equate 480 to N0 * e^(r*1) and solve for r. The initial population N0 is 60, so 480 = 60 * e^r, which simplifies to 8 = e^r. Taking the natural logarithm of both sides gives us ln(8) = r.
The relative growth rate, r, can now be calculated as follows:
- ln(8) is approximately 2.079.
Therefore, the relative growth rate of the E. coli culture is approximately 2.079 per hour.