Final answer:
To calculate the projected population of bacteria after 50 hours at a growth rate of 2.5% every 20 hours, we use the exponential growth formula, resulting in the future population size after the necessary calculations.
Step-by-step explanation:
The question involves calculating the projected population of a culture of bacteria that increases at a rate of 2.5% every 20 hours and projecting how many bacteria there will be after 50 hours from an initial count of 100 bacteria. To solve this, we'll use the formula for exponential growth:
N = N0 * e^{(r*t)}
where:
- N is the future population size,
- N0 is the initial population size,
- e is the base of the natural logarithm (approximately 2.71828),
- r is the growth rate per unit of time, and
- t is the time that has passed.
In this case, the growth rate (r) is 2.5% or 0.025 when expressed as a decimal. Every 20 hours is our unit of time, and we want to find out the population after 50 hours. Since the growth rate is given for every 20 hours, we need to adjust our time (50 hours) to match this interval. This results in t = 50/20 = 2.5 intervals of 20 hours each.
Now we can plug the values into our formula:
N = 100 * e^{(0.025*2.5)}
To find N, we simply perform the calculation:
N = 100 * e^{(0.0625)}
After performing the necessary calculations, we round the answer to an appropriate number of significant figures, which provides us with the expected number of bacteria after 50 hours. This is an illustration of population dynamics and an example of how prokaryotic fission can lead to exponential population growth under ideal conditions.