The number of bacteria in a certain population increases according to a continuous growth model, with a growth rate parameter of 25% per hour. An initial sample is obtained from this population, and after six hours, the sample has grown to 7081 bacteria. Find the number of bacteria in the initial sample. Round your answer to the nearest integer.
Note: This is a continuous exponential growth model.
And though the growth rate parameter is 25% per hour, the actual growth is not 25% each hour.
To find the number of bacteria in the initial sample, we will use the formula for continuous exponential growth:
N = N0 * e^(rt),
where:
N is the final number of bacteria,
N0 is the initial number of bacteria,
e is Euler's number (approximately 2.71828),
r is the growth rate parameter, and
t is the time in hours.
In this case, N = 7081 (the final number of bacteria), r = 0.25 (the growth rate parameter), and t = 6 hours.
To find N0, we can rearrange the formula as follows:
N = N0 * e^(rt)
N / e^(rt) = N0
Now we can substitute the given values and calculate N0:
N0 = 7081 / e^(0.25 * 6)
N0 ≈ 7081 / e^1.5
Using a calculator, we can find the approximate value of e^1.5 ≈ 4.48169.
Therefore, N0 ≈ 7081 / 4.48169 ≈ 1579.22.
Rounding to the nearest integer, the number of bacteria in the initial sample is approximately 1579.