3 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_520_873_478_675}
\captionsetup{labelformat=empty}
\caption{Fig. 12.1}
\end{figure}
- State the value of \(x\) at A , where the arch meets the road.
- Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph.
Hence find the height of the arch.
- Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717}
\captionsetup{labelformat=empty}
\caption{Fig. 12.2}
\end{figure} - Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.