Final answer:
The given problem is a variant of exponential growth known as logistic growth, commonly seen in bacteria populations. However, the actual growth rate in this scenario varies due to the cosine term in the differential equation which needs advanced methods for it to be solved. Cell populations often show exponential growth under ideal conditions.
Step-by-step explanation:
In the given question, we have a differential equation representing the rate at which a cell population may be growing. We're given information about the number of cells at time t=0 and need to find the number of cells at time t=24. The given differential equation is dN/dt = 2(1 - cos(2πt/24))N, which is a logistic growth model, a form of exponential growth wherein the growth rate is proportional to the current population's size, but decreases as the population nears its carrying capacity.
Bacterial cells often exhibit an exponential growth pattern. If we had a simple exponential growth scenario—for instance, if a single cell divided every 30 minutes—we could calculate the number of cells after 24 hours using the formula 2^n, where 'n' is the number of divisions. Here, 'n' would be 48 (since 24 hours = 48 half-hours). This would yield around 2.8 × 10¹⁴ cells at the end of the day.
However, in the given problem, the growth rate varies due to the cosine term in the differential equation. This means the population growth over the day is not as straight-forward as in our simpler exponential growth example. We would need to solve the given differential equation with the provided initial condition (N(0) = 3) to get the number of cells at t = 24 hours. Unfortunately, this might need more advanced mathematical methods, like numerical approximation or the application of Fourier analysis.
Learn more about Exponential Growth