Circular arc problems

12 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm , held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are \(A , B , C , D\), \(E\) and \(F\). Points \(P\) and \(Q\) are situated where straight sections of the rope meet the pipe with centre \(A\).

1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
2. Find the length of the rope.
3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
4. Find the area of the complete region enclosed by the rope.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .

1. Show that \(r = 2\)
2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.

8. \begin{figure}[h] \end{figure} Figure 2 shows the cross section \(A B C D\) of a small shed. The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m . The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line. Given that the size of \(\angle B A C\) is 0.65 radians, find

1. the length of the arc \(B C\), in m , to 2 decimal places,
2. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
3. the size of \(\angle C A D\), in radians, to 2 decimal places,
4. the area of the cross section \(A B C D\) of the shed, in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

13 Fig. 13.1 shows a greenhouse which is built against a wall. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The greenhouse is a prism of length 5.5 m . The curve AC is an arc of a circle with centre B and radius 2.1 m , as shown in Fig. 13.2. The sector angle ABC is 1.8 radians and ABD is a straight line. The curved surface of the greenhouse is covered in polythene.

1. Find the length of the arc AC and hence find the area of polythene required for the curved surface of the greenhouse.
2. Calculate the length BD .
3. Calculate the volume of the greenhouse.

7. \begin{figure}[h] \end{figure} Fig. 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates (7,24), (21,24) and (28,0) respectively.

1. Show that the length of \(B M\) is 25 mm .
2. Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
3. Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
4. calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.