How do you model population growth using an exponential function?

You model population growth using an exponential function by using the formula \( P(t) = P_0 e^{rt} \).

In this formula, \( P(t) \) represents the population at time \( t \), \( P_0 \) is the initial population size, \( e \) is the base of the natural logarithm (approximately 2.71828), \( r \) is the growth rate, and \( t \) is the time period over which the population grows. This formula assumes that the population grows continuously and proportionally to its current size.

To break it down, let's say you start with an initial population \( P_0 \) of 1000 people, and the population grows at a rate \( r \) of 5% per year. You would convert the percentage growth rate to a decimal by dividing by 100, so \( r = 0.05 \). If you want to find the population after 3 years, you would substitute these values into the formula:

\[ P(3) = 1000 \times e^{0.05 \times 3} \]

Using a calculator to find \( e^{0.15} \), you get approximately 1.1618. So,

\[ P(3) = 1000 \times 1.1618 = 1161.8 \]

This means the population would be approximately 1162 people after 3 years.

Exponential growth is characterised by the fact that the rate of growth is proportional to the current population. This results in a rapid increase over time, which is why it's called "exponential." Understanding this concept is crucial for analysing real-world scenarios like population growth, where resources and space can become limiting factors.