Exponential and Logarithmic Functions - Applications of Exponential and Logarithmic Functions

The stated population 1226 is rounded to the nearest animal. Treat the given time of the maximum as exact and the supplied count as approximate.

The population of a gold-mining town, \(N\), initially grew over time, reaching a maximum of \(1226\) after \(3\frac{1}{3}\) years. However, as the gold supply was depleted, the population began to decline.

The population at any time \(t\) (in years) is modelled by the equation:\(N=1000t{{\mathrm{e}}^{-at}}\), where \(a\) is a constant. The diagram illustrates the behaviour of the population over time.

1. Calculate the population of the town at \(t=0\).
2. Find the population of the town when \(t=2\).
3. Describe the behaviour of the population as \(t\) increases indefinitely.