8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_499_551_246_758}
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\caption{Figure 4}
\end{figure}
Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground.
The vertical height, \(h\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit.
The ball travels a horizontal distance of \(d\) metres before it first hits the ground.
The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
The path of the ball is modelled by the equation
$$h = 1.5 x - 0.5 x \mathrm { e } ^ { 0.02 x } \quad 0 \leqslant x \leqslant d$$
\section*{Use the model to answer parts (a), (b) and (c).}
- Find the value of \(d\), giving your answer to 2 decimal places.
(Solutions relying entirely on calculator technology are not acceptable.) - Show that the maximum value of \(h\) occurs when
$$x = 50 \ln \left( \frac { 150 } { x + 50 } \right)$$
Using the iteration formula
$$x _ { n + 1 } = 50 \ln \left( \frac { 150 } { x _ { n } + 50 } \right) \quad \text { with } x _ { 1 } = 30$$
- find the value of \(x _ { 2 }\) to 2 decimal places,
- find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_2270_56_309_1981}