Final answer:
To find the initial bacterial population given a population of 70,000 at 80 minutes and a doubling period of 15 minutes, divide 80 by 15 to get the number of doubling periods, and use the exponential growth formula to solve for the initial population, which results in approximately 4,108 bacteria.
Step-by-step explanation:
The student's question pertains to the concept of exponential growth and how to find the initial population of bacteria given the population at a later time and the doubling period. Since the doubling period is 15 minutes, we need to calculate how many doubling periods have occurred by the time 80 minutes have passed, and then use this information to find the initial population.
First, we calculate the number of doubling periods by dividing 80 minutes by the doubling period of 15 minutes:
Number of doubling periods = 80 minutes / 15 minutes per period = 5.33 periods (approximately).
Since the population doubles every period, we can use the formula for exponential growth N = N0 * 2^x, where N is the final population size, N0 is the initial population size, and x is the number of doubling periods. Plugging in the values we have:
70000 = N0 * 2^5.33.
To solve for N0, we divide both sides by 2^5.33:
Initial population (N0) = 70000 / 2^5.33.
Using a calculator, we find that the initial population is approximately 4108 bacteria.