Question Bank | Unit 5B - Trigonometry - Bearing Problems, Angle of Elevation and Depression, 3 Dimensional Problems

1. A ship sails directly from a port $P$ to a point $Q$ , which is $10 nautical miles$ due east of a lighthouse $L$ . Given that $P \overset{\land}{L} Q = 70^{°}$ and $P \overset{\land}{Q} L = 30^{°}$ , calculate

(a) the bearing of $Q$ from $P$ ,

(b) the bearing of $L$ from $P$ ,

(c) the distance $P Q$ ,

(d) the shortest distance between the lighthouse and the ship during its journey.

2. The diagram shows a pyramid with a rectangular base $A B C D$ and vertex $V$ . The slant edges $V A$ , $V B$ , $V C$ and $V D$ are all equal and the diagonals of the base intersect at $N$ . $A B = 8 cm$ , $A C = 10 cm$ and $V N = 12 cm$ .

(a) Calculate $B C$ .

(b) Calculate $V C$ .

(c) Write down the tangent of the angle between $V N$ and $V C$ .

3. In the diagram, $L$ is $5 km$ due north of $K$ , $L M = 3 km$ and $K \overset{\land}{L} M = 120^{°}$ . Calculate

(a) the bearing of $M$ from $L$ ,

(b) the distance that $L$ is east of $M$ ,

(c) the distance $K M$ .

4. $A$ , $B$ , $C$ and $D$ are four points on horizontal ground. $C D = 140 m$ , $A D = 100 m$ , $B$ is $50 m$ due south of $A$ and $C$ is $60 m$ due east of $A$ . Calculate

(a) $A \overset{\land}{B} C$ ,

(b) $D \overset{\land}{A} C$ ,

(c) the bearing of $A$ from $D$ .

A vertical mast stands at the point $A$ and the angle of elevation of the top of this mast from each of the points $B$ , $C$ and $D$ is known. Given that the smallest of these angles of elevation is $12^{°}$ , calculate the height of the mast in metres, giving your answer correct to three significant figures.

5. $A B C$ are three points on level ground. $B$ is $80 m$ due east of $C$ and $B C = B A$ . The point $P$ on $A B$ is such that $B P = 30 m$ and $C P = 60 m$ .

(a) Calculate

(i) $C \overset{\land}{B} P$ ,

(ii) the bearing of $A from C$ .

(b) A vertical mast stands at the point $B$ and the angle of elevation of the top of the mast from $A$ is $10^{°}$ . Calculate the height of the mast.

6. The towns $P$ , $Q$ and $R$ are such that $P$ and $Q$ are equidistant from $R$ . The bearing of $P$ from $R$ is $100^{°}$ and the bearing of $Q$ from $R$ is $220^{°}$ . Calculate the bearing of

(a) $R$ from $Q$ ,

(b) $Q$ from $P$ .

7. $G$ , $H$ , $K$ and $L$ are four points on level ground. $G K = 4 m$ , $K L = 5 m$ , $G L = 7 m$ , $H \overset{\land}{G} K = 47^{°}$ and $L \overset{\land}{G} K = 70^{°}$ .

(a) Calculate

(i) $H K$ ,

(ii) $G \overset{\land}{K} L$ .

(b) A vertical pole, whose top is $T$ , is erected at the midpoint of $G L$ . Given that the height of the pole is $2·87 m$ , calculate $T \overset{\land}{G} L$ .

8. $P$ , $Q$ , $R$ and $S$ are the four corners of a rectangular plot marked out on level ground. Given that the bearing of $Q$ from $P$ is $020^{°}$ and the bearing of $R$ from $P$ is $090^{°}$ , calculate the bearing of

(a) $P$ from $Q$ ,

(b) $R$ from $Q$ ,

(c) $S$ from $Q$ .

9. Triangle $P Q R$ is in a vertical plane, $P Q$ is inclined at $30^{°}$ to the horizontal and $P \overset{\land}{R} Q = 95^{°}$ . Given that the angle of elevation of $R$ from $P$ is $75^{°}$ , calculate

(a) $R \overset{\land}{P} Q$ ,

(b) the angle of elevation of $R$ from $Q$ .

10. $D$ , $E$ and $F$ are three points on level ground. The bearing of $E$ from $D$ is $049^{°}$ and $D \overset{\land}{E} F = 106^{°}$ . Calculate the bearing of $F from E$ .

11. (a) In the triangle $P Q R$ , $P Q = 8.9 cm$ , $∠ R = 54^{°}$ and $∠ P = 76^{°}$ . Calculate the length of $Q R$ .

(b) (i) A triangular frame $B D C$ is such that $B D = 6 m$ , $C D = 5 m$ and $∠ B D C = 110^{°}$ . Calculate the length of $B C$ .

(ii) The frame $B D C$ is part of the structure of a roof. $D$ is vertically above a point $A$ which is in the same horizontal plane as $B$ and $C$ . Given that $∠ A B D = 17^{°} 42^{'}$ , calculate $∠ A C D$ .

12. In the diagram $F$ , $G$ and $H$ represent three points on a map, calculate

(a) the bearing of $G$ from $F$ ,

(b) the bearing of $F$ from $H$ .

13. The diagram shows three points $A$ , $B$ and $C$ on level ground at the corners of an equilateral triangle. Given that the bearing of $B$ from $A$ is $160^{°}$ , calculate the bearing of

(a) $C$ from $A$ ,

(b) $A$ from $B$ ,

(c) $B$ from $C$ .

14. In the diagram, $A$ is $10 km$ due north of $C$ . $A B = 5 km$ , $B C = x km$ and $∠ C A B = 70^{°}$ .

(a) Calculate the bearing of $A$ from $B$ .

(b) Using as much of the information below as is necessary, calculate the value of $x^{2}$ .

[ $\sin 70^{°} = 0.940$ , $\cos 70^{°} = 0.342$ , $\tan 70^{°} = 2.75$ ]

15. $A$ , $B$ , $C$ and $D$ are four points on level ground with $B$ due east of $A$ . It is given that $A C = 55 m$ , $C D = 25 m$ , $A D = 70 m$ , $∠ C A B = 50^{°}$ and $∠ A B C = 48^{°}$ .

(a) Calculate

(i) $A B$ ,

(ii) $∠ C A D$ ,

(iii) the bearing of $D$ from $A$ .

(b) A man walks due east from $A$ until he reaches a point $P$ which is equidistant from $A$ and $C$ . Calculate the distance $A P$ .

16. (a) A boat is sailing at a steady speed of $18 knots$ . Calculate, in nautical miles, the distance it will travel in $2 \frac{1}{4} hours$ .

(b) Another boat is sailing on a bearing of $019^{°}$ and the captain orders the course to be changed by $24^{°}$ . Find the two possible bearings on which the boat might then sail.

17. A pyramid has a horizontal square base $A B C D$ of side $6 cm$ whose diagonals intersect at $X$ . The vertex $V$ is $7 cm$ vertically above $X$ . The midpoint of $C D$ is $M$ . Calculate

(a) $V M$ ,

(b) $∠ B V X$ .

18. Given that $M$ , $A$ , $T$ , $B$ and $O$ represent points on the ground, and that $T$ is due east of O, find the bearing of $A$ from $T$ .

19. In the diagram, $A$ is north of $C$ , $∠ B A C = 30^{°}$ , $∠ A C B = 74^{°}$ and $B C = 4 km$ .

(a) Find the bearing of $B$ from $A$ .

(b) Given that $\sin 30^{°} = 0.50$ and $\sin 74^{°} = 0.96$ , find $A B$ .

20. Three points $S$ , $Y$ and $E$ , are on level ground, $Y$ is the foot of a vertical flagpole $X$ $Y$ . $S$ is $40 m$ due south of $Y$ and $E$ is due east of $Y$ .

(a) The angle of elevation of $X$ from $S$ is $15^{°}$ . Calculate the height of the flagpole.

(b) The bearing of $E$ from $S$ is $037^{°}$ . Calculate the distance $E S$ .

(c) Calculate the shortest distance from $Y$ to the line $E S$ .

(d) Calculate the angle of elevation of $X$ from $E$ .

21. In the diagram, $C$ represents the foot of a vertical tower $C T$ . The points $A$ , $B$ and $C$ are on horizontal ground, where $∠ C A B = 90^{°}$ , $A C = 70 m$ and $B C = 100 m$ . Given that the angle of elevation of $T$ from $A$ is $26^{°}$ calculate

(a) the height of the tower,

(b) $∠ A B C$ ,

(c) the angle of depression of $B$ from $T$ .

22. A ship sails $8 km$ from $P$ to $Q$ . It then sails $5 km$ from $Q$ to $R$ on a bearing of $075^{°}$ .

(a) Given that $∠ P Q R = 140^{°}$ , calculate

(i) the bearing of $Q$ from $P$ ,

(ii) how far $Q$ is east of $P$ ,

(iii) the distance $P R$ .

(b) The ship finally sails from $R$ to a position $T$ , which is due north of $Q$ . Given that $∠ Q T R = 40^{°}$ , calculate

(i) the distance $R T$ ,

(ii) the shortest distance between the point $Q$ and the ship as it sails from $R$ to $T$ .

23. A ship leaves a port $P$ and sails for $5 km$ on a bearing of $120^{°}$ to a port $Q$ .

(a) Find the bearing of $P$ from $Q$ .

(b) Using as much of the information given below as is necessary, find how far $Q$ is east of $P$ .

[ $\sin 30^{°} = 0.500$ , $\cos 30^{°} = 0.866$ , $tan30^{°}=0.577$ ]

[ $\sin 60^{°} = 0.866$ , $\cos 60^{°} = 0.500$ , $tan60^{°}=1.732$ ]

24. $P$ , $Q$ , $R$ and $S$ are points on level ground, with $P$ north of $S$ . $∠ S P Q = 40^{°}$ , $∠ S Q P = 62^{°}$ , $S Q = 5 km$ and $Q R = 11 km$ .

(a) Calculate $P S$ .

(b) Calculate $R S$ .

(c) Given that $X$ is the point on $Q R$ such that $Q X = 5 km$ , calculate

(i) the shortest distance from $Q$ to $S X$ ,

(ii) the bearing of $S$ from $X$ .

25. The triangle $A O B$ lies in a horizontal plane and $T$ is vertically above $O$ . $O A = 8 m$ , $O B = 11 m$ , $O T = 6 m$ , $A T = 10 m$ and $∠ A O B = 90^{°}$ .

(a) Write down the value of $\cos ∠ A T O$ , giving your answer as a fraction.

(b) Calculate the volume of the pyramid $O A B T$ .

[Volume of a pyramid = $\frac{1}{3}$ area of base × perpendicular height]

26. (a) A surveyor is carrying out a survey on horizontal ground. From a point $O$ she observes a point $A$ which is $80 m$ from $O$ on a bearing of $040^{°}$ . The surveyor also observes a point $B$ which is $90 m$ from $O$ and due south of the point $A$ . Calculate

(i) the bearing of $O$ from $A$ ,

(ii) the angle $O B A$ .

(b) The point $C$ is $100 m$ due west of $O$ . The surveyor walks directly from $C$ to $A$ . How far does she walk?

(c) The surveyor then walks from $A$ towards $B$ until she reaches a point $P$ , where $C P$ is a minimum. Calculate $C P$ .

27. In the diagram, the bearing of $B$ from $A$ is $064^{°}$ , $∠ A C B = 70^{°}$ and $A B = B C$ . Calculate

(a) the bearing of $A$ from $B$ ,

(b) $∠ A B C$ ,

(c) the bearing of $C$ from $B$ .

28. The diagram represents a triangular prism in which three of the faces are rectangular. Given that $B E = 30 cm$ , $A B = 10 cm$ , $A C = 8 cm$ and $∠ B A C = 47^{°}$ , use as much of the information given below as is necessary to calculate

(a) the area of $Δ A B C$ ,

(b) the volume of the prism.

[ $\sin 47^{°} = 0.731$ , $\cos 47^{°} = 0.682$ , $\tan 47^{°} = 1.072$ ]

29. $A B C D$ represents a building with a vertical flagpole $A P$ on the roof. The point $O$ is on the same level as $C$ and $D$ . The angle of elevation of $A$ from $O$ is $15^{°}$ , $O A = 60 metres$ and $∠ P O A = 7^{°}$ .

(a) Calculate

(i) the height $A D$ of the building,

(ii) the height of the flagpole, $A P$ .

(b) Given also that $A B = 10 metres$ , calculate the angle of elevation of $P$ from $B$ .

30. (a) In the triangle $G H K$ , $G H = 9 cm$ , $H K = 10 cm$ and $K G = 6 cm$ . Calculate $∠ K G H$ .

(b) $A B C D$ represents the rectangular sloping surface of a desk. $A B E F$ is rectangle which is horizontal, and $C E$ and $D F$ are vertical lines.

$A B = D C = F E = 40 cm$ , $B C = A D = 30 cm$ , $∠ C B E = ∠ D A F = 35^{°}$ Calculate

(i) $A C$ ,

(ii) $C E$ ,

(iii) $∠ F A E$ .

31. Ann stands at $A$ , which is at the top of a vertical cliff $A C$ . She sees a boat at $B$ , which is $80 m$ from $C$ . The angle of depression of $B$ from $A$ is $35^{°}$ .

(a) Using as much of the information given below as is necessary, calculate the height of the cliff.

(b) A yacht is on the lake at $D$ , where $C D B$ is a straight line. The angle of elevation of $A$ from $D$ is $55^{°}$ . Calculate the distance $B D$ .

[ $\sin 35^{°} = 0.574$ $\cos 35^{°} = 0.819$ $\tan 35^{°} = 0.700$ ]

32. In the diagram, $A$ , $B$ and $C$ represent three islands. $B$ is $25 km$ from $A$ on a bearing of $020^{°}$ . $C$ is $36 km$ from $A$ on a bearing of $075^{°}$ .

(a) Calculate the bearing of $A from B$ .

(b) Calculate the distance of $C from B$ .

(c) A ship leaves $A$ at 12 40 and sails, directly to $C$ at a steady speed of $20 km/h$ .

(i) When the ship is at $D$ , it is due south of $B$ . Calculate the distance $A D$ .

(ii) Find the time, to the nearest minute at which the ship is closest to $B$ .

33. Villages $B$ and $C$ are each 5 kilometres from village $A$ , and $B \overset{\land}{A} C = 150^{°}$ . The village $C$ is due south of a point $X$ and the villages $A$ and $B$ are both due east of $X$ .

(a) Calculate the bearing of $C from A$ .

(b) Calculate the bearing of $B from C$ .

(c) Using as much of the information given below as is necessary, calculate

(i) how far $C$ is west of $A$ ,

(ii) the area of the triangle $A B C$ .

$[\sin 30^{°} = 0·5 30^{°} = 0·8660 \tan 30^{°} = 0·5774]$

34. $P$ , $Q$ and $R$ are three points on level ground with $P$ due north of $R$ . Angle $Q \overset{\land}{P} R = 40^{°}$ and $P Q = P R$ . Calculate the bearing of

(a) $Q from P$ ,

(b) $P from Q$ ,

(c) $Q from R$ .

35. Points $A$ , $B$ , $C$ and $D$ lie on level ground. The point $D$ is due north of $A$ . $D \overset{\land}{A} C = 140^{°}$ , $C \overset{\land}{A} B = 90^{°}$ and $A \overset{\land}{B} C = 75^{°}$ .

Find the bearing of

(a) $A from C$ ,

(b) $B from A$ ,

(c) $C from B$ .

36. In the diagram, the rectangle $K L M N$ represents a vertical cliff and $K N B R$ represents part of the horizontal surface of the sea. The sea meets the cliff along the horizontal line $K N$ . A boat is at the point $B$ , a rock is at the point $R$ and $R B N$ a horizontal straight line. $L K = 56 m$ , $K N = 80 m$ , $N B = 50 m$ , $K \overset{\land}{N} R = 60^{°}$ and $K \overset{\land}{R} N = 52^{°}$ .

(a) Calculate

(i) $K R$ ,

(ii) $K B$ .

(b) A man stands at $L$ . Calculate his angle of elevation from $B$ .

(c) The man walks along the top of the cliff from $L to M$ . Calculate his maximum angle of elevation from $B$ during this walk.

37. The bearing of $B$ from $A$ is $031^{°}$ , the bearing of $C$ from $A$ is $129^{°}$ and $A B = A C$ . Calculate

(a) the bearing of $A from B$ ,

(b) angle $A \overset{\land}{C} B$ ,

(c) the bearing of $C from B$ .

38. $P A B C D$ is a pyramid standing on a horizontal base $A B C D$ . $A B C D$ is a square with sides of length $6 cm$ . $E$ is the midpoint of $A D$ . $P E$ is vertical and $P E = 6 cm$ . Calculate

(a) the volume of $P A B C D$ ,

(b) $P A^{2}$ ,

(c) $P B$ .

$[The volume of a pyramid = \frac{1}{3} \times base area × height.]$

39. The bearing of $C$ from $A$ is $076^{°}$ , the bearing of $B$ from $A$ is $118^{°}$ and $C A = C B$ . Calculate

(a) the bearing of $A from C$ ,

(b) $A \overset{\land}{C} B$ ,

(c) the bearing of $B from C$ .

40. The diagram shows some beams which support the roof of a house. $A D C$ is a straight line, $A D = 2·8 m$ , $D B = 2·4 m$ , $A \overset{\land}{D} B = 117^{°}$ and $D \overset{\land}{B} C = 44^{°}$ .

(a) Calculate

(i) $A B$ ,

(ii) $D C$ .

(b) An extra beam is to be added to join the midpoint of $B D$ to a point on $B C$ . Calculate the length of this extra beam if it is

(i) parallel to $D C$ ,

(ii) perpendicular to $B C$ .

41. In a competition, the competitors follow a course $O A B C$ , as indicated on the diagram. $A$ is $1000 m$ due North of $O$ . $B$ is $1200 m$ from $A$ on a bearing of $120^{°}$ . $C$ is $1100 m$ from $B$ and $A \overset{\land}{B} C = 90^{°}$ .

(a) Using a scale of $1 cm to represent 100 m$ , make an accurate scale drawing of the course.

[You should place the point $O$ half way down the left hand side of a new page.]

(b) Use your drawing to find the bearing of $O from C$ .

(c) Competitors are told that they have to go from $C to a point X$ .

The point $X$ is within the quadrilateral $O A B C$ , is $600 m from C$ and is equidistant from $A O$ and $A B$ .

(i) On your scale drawing construct the locus of points which are within the quadrilateral $O A B C$ and

(I) $600 m from C$ ,

(II) equidistant from $A O$ and $A B$ .

(ii) Mark clearly the position of $X$ .

42. The diagram represents a map showing a harbour $H$ and three oil rigs, $P$ , $Q$ and $R$ , where $R$ is due East of $H$ . $H P Q$ is a straight line which lies on a bearing of $060^{°}$ and angle $H \overset{\land}{P} R = 104^{°}$ . It is given that $H P = 59 km$ , $P R = 41 km$ and $R Q = 53 km$ .

(a) A supply ship leaves $P$ at 10 45. It sails directly to $R$ , where it stays for 50 minutes, then goes on to $Q$ .

When moving it may be assumed that it travels at a constant speed of $12 km/h$ . At what time does it arrive at $Q$ ?

(b) Calculate the distance $H R$ .

(c) Calculate the bearing of $R from Q$ .

43. The diagram shows the positions of three towns $X$ , $Y$ and $Z$ , on a map which is drawn to a scale of $1 cm to 50 km$ .

(a) By making appropriate measurements, find

(i) the actual distance, in kilometres, of town $Z from town X$ ,

(ii) the bearing of $Z from X$ .

(b) Planners decide to build another town, within the triangle $X Y Z$ . It is to be equidistant from $X$ and $Y$ and not more than $300 km from Z$ .

By making appropriate constructions, indicate clearly, on the diagram, possible positions of the new town.

44. The diagram in the answer space is a map drawn to a scale of $1 cm to 5 km$ . The points $A$ , $B$ and $C$ on the map represent three places $A$ , $B$ and $C$ which are joined by straight roads.

It is known that some treasure is buried at $P$ , inside triangle $A B C$ , where $P$ is equidistant from $C A$ and $C B$ and $12 km from A B$ .

(a) By making appropriate constructions on the diagram indicate clearly the position of $P$ , where the treasure is buried.

(b) Use the diagram to find

(i) the distance, in kilometres, from $A to P$ ,

(ii) the bearing of $P from A$ .

(Note: The map drawn here has been scaled down 50% from the original diagram.)

45. A man, who is $1·94 m$ tall, stands on horizontal ground $40 m$ from a tree. The angle of elevation of the top of the tree from his eyes is $33^{°}$ . Use as much of the information below as is necessary to calculate an estimate of the height of the tree. Give your answer to a reasonable degree of accuracy.

$[\sin 33^{°} = 0·545, \cos 33^{°} = 0·839, \tan 33^{°} = 0·649]$

46. After an accident at sea, a search for survivors is carried out in the triangular region $A B C$ shown in the diagram.

$A B = 30·7 km$ , $A C = 21·2 km$ , $B$ is due north of $A$ and the bearing of $C from A$ is $073^{°}$ .

(a) Calculate

(i) the area of triangle $A B C$ ,

(ii) $B C$ .

(b) A helicopter finds a life raft at a point $X$ .

The bearing of $X from A$ is $061^{°}$ .

The bearing of $X from B$ is $146^{°}$ .

A rescue boat leaves $A$ and travels directly to $X$ at a speed of $40 km/h$ .

Calculate

(i) $A X$ ,

(ii) the time, in minutes, that the boat takes to reach $X$ .

47. In the diagram, which is not drawn to scale, $A$ , $B$ , $C$ and $D$ represent four towns. $B C = 5 km$ , $C D = 6 km$ , $A \overset{\land}{C} B = 48^{°}$ , $C \overset{\land}{B} D = 70^{°}$ , $D$ is due east of $B$ and $A \overset{\land}{B} C = 90^{°}$ .

(a) Calculate

(i) the bearing of $B from A$ ,

(ii) the distance $A C$ ,

(iii) the angle $B \overset{\land}{D} C$ .

(b) A map of this area is drawn to a scale of $1 cm to 5 km$ .

(i) Calculate the distance, in centimetres, between the points representing $C$ and $D$ on the map.

(ii) A forest is represented by an area of $3 {cm}^{2}$ on the map. Calculate the actual area, in square kilometres, of the forest.

48. In the diagram, $A$ , $B$ , $C$ and $D$ are four markers on a horizontal field. $B D = 15 m$ , $D C = 12 m$ , $A \overset{\land}{B} D = 35^{°}$ , $A \overset{\land}{D} B = 90^{°}$ and $B \overset{\land}{D} C = 160^{°}$ .

A vertical pole of height $10 m$ is positioned at $D$ . Calculate the angle of elevation of the top of the pole from $C$ .

49. The bearing of $B from A$ is $072^{°}$ .

(a) Find the bearing of $A from B$ .

(b) $C$ is due South of $B$ and $B A = B C$ .

Find the bearing of $A from C$ .

50. In the diagram, $A B C$ represents a horizontal triangular field and $A D$ represents a vertical tree in the corner of the field. A path runs along the edge $B C$ of the field $A B = 83 m$ , $A C = 46 m$ and angle $B \overset{\land}{A} C = 67^{°}$ .

(a) The angle of elevation of the top of the tree when viewed from $B$ is $14^{°}$ . Calculate the height of the tree.

(b) Calculate the length of the path $B C$ .

(c) Calculate the area of the field $A B C$ .

(d) Calculate the shortest distance from $A$ to the path $B C$ .

(e) Calculate the greatest angle of elevation of the top of the tree when viewed from any point on the path.

51. The bearing of $X from Y$ is $070^{°}$ and $X Z = Y Z$ .

(a) Find the bearing of $Y from X$ .

(b) Calculate angle $X \overset{\land}{Z} Y$ .

(c) Find the bearing of $Z from X$ .

52. Two coastguard stations, $A$ and $B$ , are $150 km$ apart with $A$ due North of $B$ . The coastguards are attempting to find the position of a ship. Radio signals indicate that this ship is

Ⅰ on a bearing of $146^{°}$ from $A$ ,

Ⅱ within $100 km$ of $B$ and

Ⅲ nearer to $B$ than $A$ .

(a) Using a scale of $1 cm to 10 km$ ,

(i) mark the position of $B$ ,

(ii) draw the 3 loci, corresponding to Ⅰ, Ⅱ and Ⅲ.

(b) On your drawing label the two extreme positions of the ship, $S_{1}$ and $S_{2}$ .

(c) The bearing of the ship from $B$ is $x^{°}$ . By considering the two extreme positions, $S_{1}$ and $S_{2}$ , of the ship, copy and complete the possible statement $_______< x <_______$ .

53. Three buoys, $A$ , $B$ and $C$ , are positioned in a lake to provide a course for a yacht race.

$A B = 800 m$ , $A \overset{\land}{B} C = 32^{°}$ , $B \overset{\land}{A} C = 22^{°}$ and $N$ is the point on $A B$ which is $200 m$ from $A$ .

(a) A helicopter, $H$ , is hovering at a point vertically above $N$ .

(i) The angle of elevation of the helicopter from $A$ is $12^{°}$ . Calculate the height of the helicopter.

(ii) $P$ is the point on $A C$ which is nearest to the helicopter. Calculate the angle of elevation of the helicopter from $P$ .

54. The diagram shows $A$ , $B$ , $C$ and $D$ , the four corners of a horizontal rectangular field $A B C D$ . The corner $B$ is $82 metres from A$ on a bearing of $021^{°}$ and $C$ is $173 metres from A$ .

(a) Calculate

(i) the bearing of $A from B$ ,

(ii) $B \overset{\land}{A} C$ ,

(iii) the bearing of $C from A$ .

(b) A hot air balloon was hovering at $E$ , which is vertically above $C$ . The angle of elevation of the bottom of the balloon from $D$ was $35^{°}$ . Calculate

(i) the height of the bottom of the balloon above $C$ ,

(ii) the angle of elevation of the bottom of the balloon from $B$ .

(c) A bird is hovering at a height of $40 metres$ above the field. It spots its prey on the ground at an angle of depression of $63^{°}$ . Calculate the distance that the bird must fly to catch its prey.

55. The diagram is a plan of a triangular field $A B C$ , drawn to a scale of $1 cm to 50 m$ . A tree, $T$ , in the field is $250 m from A$ and is equidistant from $B A$ and $B C$ .

(a) By making appropriate constructions on the diagram, indicate clearly the position of $T$ .

(b) Use the diagram to find

(i) the distance, in metres, from $T to B$ ,

(ii) the bearing of $T from B$ .

56. In Figure 1, the points $A$ , $B$ , $C$ and $D$ are the centres of four spheres, each of radius $4 cm$ , which rest on a horizontal table. Each sphere touches two of the other spheres, so that $A B C D$ is a square of side $8 cm$ .

(a) Given that $N$ is the midpoint of $A C$ , explain why $A N^{2} = 32$ .

(b) A fifth sphere, with centre $E$ and radius $5 cm$ , is now placed on top of the other four spheres so that it touches each one of them, as shown in Figure 2. The centres of the five spheres form a pyramid, as shown in Figure 3.

(i) Write down the length $A E$ .

(ii) Calculate the length $E N$ .

(iii) Calculate the height of $E$ above the table.

57. $A$ , $B$ and $C$ are three points on horizontal ground. $B T$ is a vertical mast of height $20 m$ . The top of the mast is joined to $A$ and $C$ by straight wires. Angle $B \overset{\land}{C} T = 31^{°}$ .

(a) Calculate the length of the wire $C T$ .

(b) Given that $A B = 30 m$ , calculate the angle of elevation of $T from A$ .

58. Two corners, $A$ and $B$ , of a horizontal triangular field are $240 m$ apart. The diagram below is part of a scale drawing of the field.

(a) Find the scale of the drawing in the form $1 : n$ .

(b) Find the bearing of $A from B$ .

The third corner, $C$ , of the field is south of $A B$ . It is $220 m from A$ and $170 m from B$ .

(c) Using ruler and compasses only, find and label the position of $C$ on the scale drawing.

(d) A tree, $T$ , in the field is equidistant from the three corners $A$ , $B$ and $C$ .

(i) Showing your construction clearly, find and label the position of the tree.

(ii) Find the distance of the tree from the corners of the field.

59. A radio mast $A B$ , of height $20 m$ , stands at the top of a slope which is inclined at $18^{°}$ to the horizontal.

(a) The mast is supported by a wire $A C$ attached to a point $C$ on the slope, where $B C = 30 m$ . Calculate

(i) $A \overset{\land}{B} C$ ,

(ii) the length of the wire $A C$ .

(b) When the sun is in a certain position, the shadow cast by the mast lies down the slope, shown in the diagram by the line $B D$ . Given that $A \overset{\land}{D} B = 42^{°}$ , calculate

(i) the angle of elevation of the sun,

(ii) $D \overset{\land}{A} B$ ,

(iii) the length of the shadow $B D$ .

(c) The mast is supported by another wire $A F$ . The points $B$ , $E$ and $F$ lie on horizontal ground. Given that $B \overset{\land}{E} F = 90^{°}$ , $B E = 12 m$ and $E F = 15 m$ , calculate the length of the wire $A F$ .

60. The diagram shows $A$ , $B$ , $C$ and $D$ , the four corners of a horizontal rectangular field $A B C D$ . $A C = 110 m$ and $D \overset{\land}{C} A = 43^{°}$ .

(a) Calculate the length of $D C$ .

(b) $T C$ represents a vertical tree. The angle of elevation of $T$ from $A$ is $17^{°}$ . Calculate

(i) the height of the tree,

(ii) the angle of elevation of $T$ from $D$ .

61. Two ships, Alpha and Beta, left a port, $O$ , at noon. Alpha sailed at $12 km/h$ on a bearing of $054^{°}$ . Beta sailed at $16 km/h$ on a bearing of $130^{°}$ .

(a) At 4 p.m., Alpha was at $A$ and Beta was at $B$ . Calculate the distance $A B$ .

(b) When Beta had travelled a total distance of $100 km$ it stopped at $C$ .

(i) Calculate the time when Beta reached $C$ .

(ii) Alpha continued on its course until it reached the point $D$ , due north of $C$ . Calculate the distance $O D$ .

(c) An island, $R$ , is $100 km$ due south of the port, $O$ . Calculate the bearing of $R from C$ .

62. The diagram shows four towns, $A$ , $B$ , $C$ and $D$ . $C$ is $110 km$ due East of $A$ . $A \overset{\land}{B} C=90^{°}$ and $A B = 70 km$ .

(a) Calculate

(i) the distance $B C$ ,

(ii) the bearing of $B from A$ .

(b) Given that $D$ is due South of $C$ and that $C \overset{\land}{A} D = 28^{°}$ , calculate the distance $C D$ .

(c) An aircraft flies from $A to B$ , then from $B to C$ , then from $C to D$ and finally from $D to A$ . Calculate the total distance that it flies.

63. The diagram shows two horizontal triangular fields, $A B C$ and $A C D$ which are surrounded by hedges. It is given that $D A B$ is a straight line, $A C = 65 m$ , $C \overset{\land}{A} B = 60^{°}$ and $A \overset{\land}{B} C = 72^{°}$ .

(a) Calculate the length of the hedge $B C$ .

(b) The hedge $A D$ has length $84 m$ .

Calculate

(i) the area of the field $A C D$ ,

(ii) the length of the hedge $C D$ .

(c) A vertical tree is growing at $C$ . The angle of elevation of the top of the tree from $A$ is $14^{°}$ .

(i) Calculate the height of the tree.

(ii) A boy has climbed exactly half way up the tree. Calculate the angle of depression of $D$ when viewed by the boy.

64. A man at the top of a vertical cliff observes a boat at sea.

State the angle of depression of the boat from the man.

65. $A$ , $B$ and $C$ are three towns. $C$ is equidistant from $A$ and $B$ .

The bearing of $C from A$ is $132^{°}$ and $B \overset{\land}{A} C = 75^{°}$ .

Find

(a) (i) the acute angle $A \overset{\land}{C} B$ ,

(ii) the reflex angle $A \overset{\land}{C} B$ ,

(b) the bearing of $A from C$ ,

(c) the bearing of $A from B$ .

66. $P$ , $Q$ and $R$ are three towns.

$Q$ is equidistant from $P$ and $R$ .

The bearing of $R from Q$ is $168^{°}$ and $Q \overset{\land}{P} R = 57^{°}$ .

Find

(a) (i) the reflex angle $R \overset{\land}{P} Q$ ,

(ii) the acute angle $P \overset{\land}{Q} R$ ,

(b) the bearing of $Q from R$ ,

(c) the bearing of $Q from P$ .