Bearings and navigation

13. \begin{figure}[h] \end{figure} Figure 4 shows the position of two stationary boats, \(A\) and \(B\), and a port \(P\) which are assumed to be in the same horizontal plane. Boat \(A\) is 8.7 km on a bearing of \(314 ^ { \circ }\) from port \(P\).
Boat \(B\) is 3.5 km on a bearing of \(052 ^ { \circ }\) from port \(P\).

1. Show that angle \(A P B\) is \(98 ^ { \circ }\)
2. Find the distance of boat \(B\) from boat \(A\), giving your answer to one decimal place.
3. Find the bearing of boat \(B\) from boat \(A\), giving your answer to the nearest degree.

1. \begin{figure}[h] \end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).

1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.

3 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_488_604_895_769} A landmark \(L\) is observed by a surveyor from three points \(A , B\) and \(C\) on a straight horizontal road, where \(A B = B C = 200 \mathrm {~m}\). Angles \(L A B\) and \(L B A\) are \(65 ^ { \circ }\) and \(80 ^ { \circ }\) respectively (see diagram). Calculate

1. the shortest distance from \(L\) to the road,
2. the distance \(L C\).

11

1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} (A) Calculate the total length of the course for this race.
   (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} Calculate the total length of the course for this race.

10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h] \end{figure}

1. Write down the size of the angles LAB and LBA .
2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_765_757_203_764} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
2. 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} \captionsetup{labelformat=empty} \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.
   1. Calculate the area of sector ABC .
   2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
   3. 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.
      1. 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.
      2. 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.

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-4_492_1018_256_567} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
   \end{figure} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of \(048 ^ { \circ }\). At the same time, ship T is 6.3 km from L on a bearing of \(105 ^ { \circ }\), as shown in Fig. 10.1. For these positions, calculate
   (A) the distance between ships S and T ,
   (B) the bearing of S from T .
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-4_430_698_1350_573} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} Not to scale Ship S then travels at \(24 \mathrm {~km} \mathrm {~h} { } ^ { 1 }\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes.
   Hence find, in degrees, the bearing of ship S from the lighthouse at this time.

5 \includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-3_623_355_1123_897} Some walkers see a tower, \(T\), in the distance and want to know how far away it is. They take a bearing from a point \(A\) and then walk for 50 m in a straight line before taking another bearing from a point \(B\). They find that angle \(T A B\) is \(70 ^ { \circ }\) and angle \(T B A\) is \(107 ^ { \circ }\) (see diagram).

1. Find the distance of the tower from \(A\).
2. They continue walking in the same direction for another 100 m to a point \(C\), so that \(A C\) is 150 m . What is the distance of the tower from \(C\) ?
3. Find the shortest distance of the walkers from the tower as they walk from \(A\) to \(C\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-3_622_513_244_776} The diagram shows two points \(A\) and \(B\) on a straight coastline, with \(A\) being 2.4 km due north of \(B\). A stationary ship is at point \(C\), on a bearing of \(040 ^ { \circ }\) and at a distance of 2 km from \(B\).

1. Find the distance \(A C\), giving your answer correct to 3 significant figures.
2. Find the bearing of \(C\) from \(A\).
3. Find the shortest distance from the ship to the coastline.

3. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing the position of three phone masts, \(A , B\) and \(C\).
The masts are identical and their bases are assumed to lie in the same horizontal plane.
From mast \(C\)

• mast \(A\) is 8.2 km away on a bearing of \(072 ^ { \circ }\)
• mast \(B\) is 15.6 km away on a bearing of \(039 ^ { \circ }\)
  1. Find the distance between masts \(A\) and \(B\), giving your answer in km to one decimal place.

An engineer needs to travel from mast \(A\) to mast \(B\).

Give a reason why the answer to part (a) is unlikely to be an accurate value for the distance the engineer travels.

7 Two lifeboat stations, \(P\) and \(Q\), are situated on the coastline with \(Q\) being due south of \(P\). A stationary ship is at sea, at a distance of 4.8 km from \(P\) and a distance of 2.2 km from \(Q\). The ship is on a bearing of \(155 ^ { \circ }\) from \(P\).

1. Find any possible bearings of the ship from \(Q\).
2. Find the shortest distance from the ship to the line \(P Q\).
3. Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

Fig. 11.1 shows a village green which is bordered by 3 straight roads AB, BC and CA. The road AC runs due North and the measurements shown are in metres. \includegraphics{figure_1}

1. Calculate the bearing of B from C, giving your answer to the nearest 0.1°. [4]
2. Calculate the area of the village green. [2]

The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \includegraphics{figure_2} The new road is an arc of a circle with centre O and radius 130 m.

1. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures. [2] (B) Show that the area of land added to the village green is 5300 m² correct to 2 significant figures. [4]