Modeling Population Growth?

📌 CONCEPT: The population growth of an insect can be modeled using the differential equation dP/dt = kP, where P is the population at time t and k is the growth rate constant.

📐 RULE / FORMULA: The differential equation dP/dt = kP can be solved using separation of variables, giving P(t) = P0 * e^(kt), where P0 is the initial population.

💡 WORKED EXAMPLE: Given that the population doubles in 10 years and initially there were 500 insects, we can find k as follows: Since P(10) = 2 * P0, we have 2 * 500 = 500 * e^(10k). Simplifying, we get e^(10k) = 2. Taking the natural logarithm of both sides, we get 10k = ln(2), so k = ln(2)/10. Now, to find the population after 20 years, we substitute t = 20 into the equation P(t) = P0 * e^(kt), giving P(20) = 500 * e^(20 * ln(2)/10) = 500 * e^(2 * ln(2)) = 500 * (e^ln(2))^2 = 500 * 2^2 = 2000.

⚠️ COMMON MISTAKE: Students often forget to take the natural logarithm of both sides when solving for k, leading to an incorrect value of k.