Population Growth Paradox?

📌 CONCEPT: The population growth paradox refers to a situation where the population grows at a rate proportional to the square root of the current population, leading to a seemingly paradoxical result that the population does not increase indefinitely despite the increasing growth rate.

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

💡 WORKED EXAMPLE: Suppose Tommy's village has an initial population of 1000 people and is growing at a rate of 10√P people per year. To find the population after 10 years, we can separate the variables in the differential equation and integrate: ∫dP/√P = ∫10dt. Solving this, we get P = 1000(1 + 10t)^(2/3). Substituting t = 10, we get P ≈ 1000(11)^(2/3) ≈ 3091 people.

⚠️ COMMON MISTAKE: Students often forget to consider the initial population when solving the differential equation, leading to an incorrect population growth curve.