Final answer:
To determine the initial count of bacteria in a culture with constant exponential growth, we can solve two equations using the provided counts after 4 and 8 hours.
Step-by-step explanation:
The student asked how to determine the initial bacteria count when provided with two subsequent counts after 4 and 8 hours, assuming constant exponential growth. Exponential growth in the context of bacterial populations can be expressed by the formula N = N0 * e^(rt), where N is the final population size, N0 is the initial population size, r is the growth rate, and t is time.
To find the initial count, we need to solve for N0. Given that the population size was 2,560 after 4 hours and grew to 81,920 after another 4 hours (making it 8 hours in total), we can set up two equations with the same growth rate and solve for N0. By dividing the second count by the first, we eliminate the initial population and the growth rate from the equation, leaving us with an expression that relates the two known counts and the doubling time.
After setting up the equation 81,920 = N0 * e^(4r) and dividing it by 2,560 = N0 * e^(4r), it simplifies to 32 = e^(4r), allowing us to determine the rate r. With the rate known, we can plug it back into the original formula to find the initial count N0, which would yield the answer (b) 1,280.