- The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula
$$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$
Use this model to estimate
- the population of the town on \(1 { } ^ { \text {st } }\) January 2010,
- the year in which the population first exceeds 84000 .
The population in thousands, \(Q\), of another town is modelled by the formula
$$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
- Show that the value of \(t\) when \(P = Q\) is a solution of the equation
$$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right) .$$
- Use the iteration formula
$$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$
with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.