Modelling Population Growth?
The population of a certain city is growing at a rate proportional to the number of people already living there. If the current population is 1 million and is increasing at a rate of 5% per annum, find the population after 5 years, given that the initial population was 500,000 20 years ago. Assume a constant growth rate and a constant initial population in the past.
1 Answer
📌 CONCEPT: The population growth of a city can be modelled using a differential equation, where the rate of change of population is directly proportional to the current population.
📐 RULE / FORMULA: The general form of the differential equation for population growth is dP/dt = rP, where P is the population, r is the growth rate, and t is time. For exponential growth, the solution is P(t) = P0 * e^(rt), where P0 is the initial population.
💡 WORKED EXAMPLE: Given P0 = 500,000 and r = 5% = 0.05, we want to find the population after 5 years. Using the formula P(t) = P0 * e^(rt), we get P(5) = 500,000 * e^(0.05*5) ≈ 763,919.
⚠️ COMMON MISTAKE: Failing to account for the initial population or using an incorrect growth rate can lead to inaccurate population predictions.
11 Jul 26
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