Final answer:
Using the exponential growth formula, it will take approximately 45 minutes for a bacterial culture starting with 8000 bacteria and doubling every 15 minutes to reach a population of 57000 bacteria.
Step-by-step explanation:
To determine how many minutes it will take for a bacterial culture that starts with 8000 bacteria and doubles every 15 minutes to reach 57000, we need to use the formula for exponential growth, which is N = N0 * 2^(t/T), where N is the final population size, N0 is the initial population size, t is the total time, and T is the doubling time. We are given N0 = 8000, target N = 57000, and T = 15 minutes.
We can rearrange the formula to solve for t: t = (T * log(N/N0)) / log(2). Plugging in the numbers gives us: t = (15 * log(57000/8000)) / log(2). Calculating this gives us the number of minutes needed for the population to grow to 57000 bacteria.
Calculation:
t = 15 * log(57000/8000) / log(2)
t ≈ 15 * log(7.125) / log(2)
t ≈ 15 * 2.851 / 1
t ≈ 42.765
Since we're working with intervals of 15 minutes, we'd round up to the nearest interval, giving us 45 minutes for the culture to reach 57000 bacteria.