Final answer:
To answer the student's question, calculate the rate of change of the population by taking the derivative of the logistic function and evaluating it at the given time points. Then, find the time when this rate is at its maximum to determine when the population is increasing most rapidly.
Step-by-step explanation:
The student's question revolves around understanding logistic growth in the context of fish populations in a lake. The logistic growth model they've provided is p(t) = 19,000 / (1 + 18e(-t/5)). To find the rate at which the fish population is changing after 1 month and after 10 months, we need to take the derivative of the population function with respect to time, which gives us the rate of change of the population. The resulting function can then be evaluated at t = 1 and t = 10 to find the rates of change at those specific times.
In order to determine when the population is increasing most rapidly, we need to find the point at which the first derivative of the population function is at its maximum. This is typically at the inflection point of the function, which is when the second derivative is equal to zero. For a logistic function, this usually occurs at half the carrying capacity, which in this case would be when the population reaches half of 19,000, or 9,500 fish.