8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-16_522_673_248_696}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A car stops at two sets of traffic lights.
Figure 2 shows a graph of the speed of the car, \(v \mathrm {~ms} ^ { - 1 }\), as it travels between the two sets of traffic lights.
The car takes \(T\) seconds to travel between the two sets of traffic lights.
The speed of the car is modelled by the equation
$$v = ( 10 - 0.4 t ) \ln ( t + 1 ) \quad 0 \leqslant t \leqslant T$$
where \(t\) seconds is the time after the car leaves the first set of traffic lights.
According to the model,
- find the value of \(T\)
- show that the maximum speed of the car occurs when
$$t = \frac { 26 } { 1 + \ln ( t + 1 ) } - 1$$
Using the iteration formula
$$t _ { n + 1 } = \frac { 26 } { 1 + \ln \left( t _ { n } + 1 \right) } - 1$$
with \(t _ { 1 } = 7\)
- find the value of \(t _ { 3 }\) to 3 decimal places,
- find, by repeated iteration, the time taken for the car to reach maximum speed.