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.
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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. [0pt]
[5]
The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection.
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\includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-3_506_942_1703_629}
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\caption{Not to scale}
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Fig. 9.2
Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
Hence estimate the volume of earth removed when the tunnel is re-shaped.