14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-20_777_1319_315_370}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A town's population, \(P\), is modelled by the equation \(P = a b ^ { t }\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded.
The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log _ { 10 } P\) for the population over a period of 100 years.
The line \(l\) meets the vertical axis at \(( 0,5 )\) as shown. The gradient of \(l\) is \(\frac { 1 } { 200 }\).
- Write down an equation for \(l\).
- Find the value of \(a\) and the value of \(b\).
- With reference to the model interpret
- the value of the constant \(a\),
- the value of the constant \(b\).
- Find
- the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
- the number of years it takes the population to reach 200000 , according to the model.
- State two reasons why this may not be a realistic population model.