Modeling Population Growth?

📌 CONCEPT: The population growth model can be represented as a differential equation, where the rate of change of population is proportional to the square root of the current population, allowing us to use the concept of accumulation to find the total population after a given time.

📐 RULE / FORMULA: The formula for population growth can be modeled using the equation dP/dt = k√P, where P is the population at time t, and k is the constant of proportionality.

💡 WORKED EXAMPLE: To find the total population after 5 years, we first need to find the value of k. Given that the population reaches 12000 in 3 years, we can use the differential equation to find k. Substituting P = 12000 and t = 3, we get dP/dt = k√12000. Solving for k, we find k = (2000/3√3). Using this value of k, we can then find the population at t = 5 using the accumulation concept.

⚠️ COMMON MISTAKE: Students may fail to consider the initial population and the time period for which the population growth is required, leading to incorrect results.