The cross-sectional view of a tunnel is shown on the axes below. The line [AB] represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by y=-0.1x³ +0.8x², 2 ≤x≤8, relative to an origin O. 8 E D 6 4 B A 2 2 6 8 10 Point A has coordinates (2, 0), point B has coordinates (2, 2.4), and point C has coordinates (8,0). dy (a) (i) Find dr (ii) Hence find the maximum height of the tunnel. When x = 4 the height of the tunnel is 6.4m and when x = 6 the height of the tunnel is 7.2 m. These points are shown as D and E on the diagram, respectively. (b) Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel. (c) (i) Write down the integral which can be used to find the cross-sectional area of the tunnel. (ii) Hence find the cross-sectional area of the tunnel. с

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