A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve
$$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$
where \(x\) and \(y\) are horizontal and vertical distances in metres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653}
\captionsetup{labelformat=empty}
\caption{Figure 9.1}
\end{figure}
Using this model,
(A) find the greatest height of the tunnel,
(B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.