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=¡¼Ae¡½^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

The magnitude of A in the equation P = Ae^kt represents the initial population or starting point of the population growth.

If the magnitude of A is greater, it means that the initial population is larger. As a result, the population growth will also be larger as time progresses. For example, if A is 100, it implies that the population starts with 100 individuals, and if A is 200, it means that the population starts with 200 individuals. The larger the initial population, the faster the population will grow over time.

The magnitude of k in the equation P = Ae^kt represents the rate of population growth.

If the magnitude of k is larger, it means that the population grows at a faster rate. This indicates that the population will increase more rapidly over time. For example, if k is 0.5, the population will have slower growth compared to when k is 1.0. The larger the value of k, the steeper the population growth curve will be.

The sign of k in the equation P = Ae^kt represents the direction of population growth.

If k is positive, it indicates that the population is growing. In this case, as time passes, the population will increase exponentially. For example, if k is 0.5, the population will experience positive growth. On the other hand, if k is negative, it implies that the population is declining. This means that as time progresses, the population will decrease exponentially. For instance, if k is -0.5, the population will experience negative growth.

In summary, the magnitude of A influences the initial population size, while the magnitude of k determines the rate of population growth. Additionally, the sign of k indicates whether the population is growing or declining.