A scientist is observing the population of a certain species of fish in a lake. The population, P(t), after t years, is modeled by the function P(t) = 1500 / (1 + 9e^(-0.2t)). What is the instantaneous rate of change of the population when the population is 750 fish?

Correct answer: B. 30 fish/year

This is a logistic growth model. The instantaneous rate of change for a logistic function P(t) = K / (1 + Ae^(-kt)) is given by dP/dt = k * P * (1 - P/K). First, identify K (carrying capacity) = 1500 and k (growth rate) = 0.2. The question asks for the rate of change when P = 750. Substitute these values into the derivative formula: dP/dt = 0.2 * 750 * (1 - 750/1500) = 0.2 * 750 * (1 - 0.5) = 0.2 * 750 * 0.5 = 0.2 * 375 = 75 fish/year. Wait, recheck the formula. The rate of change for logistic growth is dP/dt = kP(1 - P/K). In our case, P(t) = K / (1 + Ae^(-kt)). The 'k' in the standard derivative formula is the 'k' in the exponent. So, k=0.2. At P=750, dP/dt = 0.2 * 750 * (1 - 750/1500) = 0.2 * 750 * (1 - 0.5) = 0.2 * 750 * 0.5 = 75. Let's re-evaluate the provided options and my prior math. The original solution led to 75. Let me re-calculate again. 0.2 * 750 * 0.5 = 150 * 0.5 = 75. This is option D. Let me re-read the question carefully for any missed detail. 'Instantaneous rate of change...when the population is 750 fish'. The question is indeed asking for dP/dt when P=750. Let me check the formula for logistic growth rate of change. It is indeed dP/dt = kP(1 - P/K). Let me verify the 'k' value. In P(t) = K / (1 + Ae^(-kt)), k is 0.2. So, K=1500, k=0.2, P=750. dP/dt = (0.2) * (750) * (1 - 750/1500) = 150 * (1 - 0.5) = 150 * 0.5 = 75. The answer should be 75. Given my options, I have a mismatch. Let me assume there was a typo in my initial options-answer pairing, and 75 is the intended correct answer. Or, there is a different form of the logistic growth derivative. Some sources use r_max * P * (1 - P/K). Here, r_max is usually k. Let me consider if the question implies P=K/2, which is the maximum growth rate. K/2 = 1500/2 = 750. Yes, the question is asking for the rate at the inflection point, where growth is maximal. So the calculation is correct. My answer must be D. If I am forced to choose from the given options and my calculated answer is D, I should state D and ensure my explanation matches. Let me correct the answer and explanation accordingly.