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 = Aekt where A and k are
constants.
(a) With the aid of appropriate examples, explain how the growth of P over time
would be influenced by:
(i) The magnitude of A
(ii) The magnitude of k
(iii) The sign of k

To understand how the growth of the population P in Metropia is influenced by the constants A and k in the equation P = Aekt, let's consider each of these factors separately with the help of examples.

(i) The magnitude of A:
The constant A in the equation represents the initial population, or the population at time t = 0. Therefore, the magnitude of A determines the starting point of the population growth.

For example, let's say the initial population A is 100,000. If the constant k is positive (indicating population growth), then a larger value of A, like 200,000, would mean a higher starting population. As a result, the population will grow to a larger number over time compared to a smaller initial population.

Conversely, if the constant k is negative (indicating population decline), a larger value of A will lead to a higher starting population, resulting in a larger decline in population over time compared to a smaller initial population.

(ii) The magnitude of k:
The constant k in the equation determines the rate at which the population grows or declines. It represents the proportional growth rate.

Consider the same initial population A of 100,000. If the constant k is positive, larger values of k will result in a faster population growth rate. For instance, if k = 0.05, the population will grow slowly over time. On the other hand, if k = 0.1, the population will grow at a faster rate.

Similarly, if the constant k is negative, larger absolute values of k will indicate a faster decline rate. For example, if k = -0.05, the population will decline slowly over time, whereas if k = -0.1, the population will decline more rapidly.

(iii) The sign of k:
The sign of the constant k determines whether the population is growing (positive k) or declining (negative k).

If k > 0, the population P will grow exponentially. As time increases, the population will continue to increase at an accelerating rate.

If k < 0, the population P will decline exponentially. As time increases, the decrease in population will also accelerate.

In summary, the magnitude of A affects the starting population, the magnitude of k influences the rate of growth or decline, and the sign of k determines whether the population is growing or declining.