The population P of a particular city, Metropia, is growing at a rate proportional to the current population.
The population at time t years is modelled by the equation P=Ae^kt where A and k are constants.
With the aid of appropriate examples, explain how the growth of P over time would be influenced by:
The Magnitude of A
The Magnitude of k
The sign of k
To understand how the growth of the population P is influenced by the magnitude of A, k, and the sign of k, let's analyze each factor individually with the help of examples.
1. Magnitude of A:
The constant A represents the initial population of the city at time t = 0. It determines the starting point of the population growth equation. The magnitude of A directly affects the size of the initial population. A larger value of A indicates a larger starting population, while a smaller value of A represents a smaller starting population.
For example, let's consider two cities, City A and City B, with the same k value but different magnitudes of A. If City A has A = 5000 and City B has A = 10000, City B will start with a higher population than City A. Over time, both cities will still experience proportional population growth, but City B will always have a larger population due to its higher initial value for A.
2. Magnitude of k:
The constant k determines the rate of population growth. It represents the proportionality constant between the population and time. A positive value for k indicates population growth, while a negative value signifies population decline.
The magnitude of k influences the speed at which the population grows or declines. A larger magnitude of k results in faster population growth compared to a smaller magnitude of k. This means that as k increases, the population increases rapidly, and as k decreases, the population grows more slowly or even decreases.
For instance, let's consider two cities, City X and City Y, with the same initial population (A) but different k values. City X has k = 0.02, indicating slow population growth, while City Y has k = 0.05, indicating fast population growth. Over a period of 10 years, City Y with the higher k will experience a significantly higher increase in population compared to City X.
3. Sign of k:
The sign of k determines whether the population is increasing or decreasing over time. A positive value of k indicates population growth, while a negative value represents population decline.
For example, consider a city with an initial population A = 5000 and a positive value of k. In this case, the population will increase over time because a positive k indicates growth. However, if the city has a negative value for k, the population will decline over time.
In summary, the magnitude of A determines the starting population, while the magnitude and sign of k determine the rate and direction of population growth/decline respectively. Understanding these influences can help in analyzing and predicting population trends in a given city or region.