Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_527_1474_1452_404}
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\caption{Fig. 11.2}
\end{figure}
BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
- Calculate the area of sector ABC .
- Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
- Hence calculate the area of cross-section of the rudder.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-5_695_1012_271_600}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{figure}
A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water. - Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough.
Hence estimate the volume of water in the trough.
- A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\).
Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
Hence find the volume of water in the trough as given by this model.