Population Growth Modeling?

📌 CONCEPT: A differential equation is used to model population growth when the rate of change of the population is proportional to the current population, and it can be used to find the population at a future time.

📐 RULE / FORMULA: The differential equation for population growth is dP/dt = kP, where P is the population, t is time, and k is the growth rate. The solution to this equation is P(t) = P0 * e^(kt), where P0 is the initial population.

💡 WORKED EXAMPLE: Suppose the initial population is 10000 and the growth rate is 2% per year. We can model this situation using the differential equation dP/dt = 0.02P. The solution to this equation is P(t) = 10000 * e^(0.02t). To find the population after 5 years, we substitute t = 5 into the solution: P(5) = 10000 * e^(0.02*5) = 10000 * e^0.1 ≈ 11071.03.

⚠️ COMMON MISTAKE: Students often forget to check if the growth rate is positive, which can lead to a negative population. Also, they may not use the correct units for the growth rate, which can affect the accuracy of the solution.