9 Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel.
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\caption{Fig. 9}
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With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| \(y\) | 0 | 4.0 | 4.9 | 5.0 | 4.9 | 4.0 | 0 |
The length of the tunnel is 50 m .
- Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel.
- An engineer models the height of the roof of the tunnel using the curve \(y = \frac { 5 } { 81 } \left( 108 x - 54 x ^ { 2 } + 12 x ^ { 3 } - x ^ { 4 } \right)\). This curve is symmetrical about \(x = 3\).
(A) Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel.
(B) Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel.