Master Trigonometry Applications: Solve Real-World Problems with Ease

Question 1

A bridge has a span of 100m. The bridge is supported by two towers that are 20m high. Calculate the angle of elevation from the top of one tower to the top of the other.

Accepted Answer

30°

Answer

10°

Answer

20°

Answer

40°

Question 2

In a right-angled triangle, the angles measure 30°, 60°, and 90°. If the side opposite the 30° angle is 6 cm, calculate the length of the side opposite the 60° angle.

Accepted Answer

3√3 cm

Answer

6 cm

Answer

3 cm

Answer

None of the above

Question 3

A surveyor wants to measure the distance across a lake. They stand on one side of the lake and measure the angle between the shoreline and a point directly across the lake to be 35 degrees. The surveyor then walks 100 meters along the shoreline and measures the angle to the same point across the lake again, this time finding it to be 50 degrees. What is the approximate distance across the lake?

Accepted Answer

143 meters

Answer

70 meters

Answer

200 meters

Answer

110 meters

Question 4

A tree casts a shadow that is 18 meters long when the angle of elevation of the sun is 30 degrees. What is the approximate height of the tree?

Accepted Answer

10.4 meters

Answer

9.0 meters

Answer

18.0 meters

Answer

6.0 meters

Question 5

A boat travels at a speed of 10 km/h on a bearing of 045 degrees. After 2 hours, how far north has the boat traveled? (Round to one decimal place.)

Accepted Answer

14.1 kilometers

Answer

20.0 kilometers

Answer

7.1 kilometers

Answer

10.0 kilometers

Question 6

A hiker walks 5 kilometers east and then 3 kilometers north. What is the approximate bearing of the hiker from their starting point?

Accepted Answer

31°

Answer

59°

Answer

121°

Answer

045°

Question 7

A building casts a shadow 120 meters long. The angle of elevation from the end of the shadow to the top of the building is 60 degrees. What is the height of the building?

Accepted Answer

120√3 m

Answer

240√3 m

Answer

60√3 m

Answer

120√2 m

Question 8

A ladder is leaning against a wall. The base of the ladder is 3 meters from the wall, and the top of the ladder reaches a height of 4 meters. What is the length of the ladder?

Accepted Answer

5 m

Answer

3 m

Answer

2 m

Answer

4 m

Question 9

In a right-angled triangle, the hypotenuse is 20 cm, and one of the angles is 30 degrees. What is the length of the side opposite the 30-degree angle?

Accepted Answer

10 cm

Answer

15 cm

Answer

25 cm

Answer

5 cm

Question 10

A ship sails due north at a speed of 20 knots. After 3 hours, it changes course by 60 degrees to the east. How far from its starting point will the ship be after a total of 5 hours? (Assume 1 knot = 1.852 km/h)

Accepted Answer

100√3 km

Answer

150√3 km

Answer

75√3 km

Answer

50√3 km

Question 11

A rectangular plot of land is 50 meters long and 30 meters wide. A diagonal path is constructed from one corner to the opposite corner. What is the approximate length of the diagonal path?

Accepted Answer

58.31 m

Answer

46.31 m

Answer

68.31 m

Answer

38.31 m

Question 12

Two ships leave a port at the same time. The first ship sails due east at a speed of 15 knots, and the second ship sails due north at a speed of 12 knots. After 4 hours, what is the approximate distance between the two ships? (Assume 1 knot = 1.852 km/h)

Accepted Answer

60√5 km

Answer

70√5 km

Answer

50√5 km

Answer

40√5 km

Question 13

A tower has a height of 50 m. From a point on the ground 100 m from the base of the tower, the angle of elevation to the top of the tower is measured to be 26.57 degrees. What is the distance from the ground to the top of the tower?

Accepted Answer

52.26 m

Answer

37.74 m

Answer

62.26 m

Answer

47.74 m

Question 14

A bridge is supported by two vertical cables that are attached to the top of two towers. The towers are 100 m apart and the cables are each 50 m long. What is the height of the towers?

Accepted Answer

75 m

Answer

25 m

Answer

50 m

Answer

100 m

Question 15

A farmer has a rectangular field that is 500 m long and 200 m wide. He wants to fence the perimeter of the field, but he leaves a 50 m wide opening in the middle of one of the longer sides for a gate. How many meters of fencing does the farmer need?

Accepted Answer

1200 m

Answer

1000 m

Answer

1400 m

Answer

800 m

Question 16

A surveyor standing 50m from the base of a vertical cliff measures the angle of elevation to the top of the cliff to be 35°. What is the height of the cliff, correct to the nearest meter?

Accepted Answer

35m

Answer

57m

Answer

65m

Answer

41m

Question 17

Two boats leave a harbor at the same time. Boat A travels at 12km/h on a bearing of 065°, and boat B travels at 15km/h on a bearing of 115°. How far apart are the boats after 2 hours?

Accepted Answer

36km

Answer

27km

Answer

39km

Answer

30km

Question 18

A hot air balloon is vertically above a point A on the ground. From a point B on the ground, 150m away from A, the angle of elevation to the balloon is 62°. What is the height of the balloon above the ground?

Accepted Answer

283m

Answer

168m

Answer

133m

Answer

266m

Question 19

A ladder 6m long leans against a vertical wall. The foot of the ladder is 2m from the base of the wall. What angle does the ladder make with the ground?

Accepted Answer

71°

Answer

27°

Answer

19°

Answer

63°

Question 20

A flagpole stands on top of a building. From a point on the ground, the angle of elevation to the top of the flagpole is 42° and the angle of elevation to the bottom of the flagpole is 38°. If the distance from the point on the ground to the base of the building is 50m, what is the height of the flagpole?

Accepted Answer

5.5m

Answer

9.6m

Answer

44.5m

Answer

4.1m

Question 21

Two ships leave a port at the same time. The first ship sails on a bearing of 130° at a speed of 20km/h. The second ship sails on a bearing of 220° at a speed of 25km/h. How far apart are the ships after 3 hours?

Accepted Answer

113km

Answer

75km

Answer

95km

Answer

105km

Question 22

From the top of a lighthouse 80m high, a boat is observed at an angle of depression of 12°. How far is the boat from the base of the lighthouse?

Accepted Answer

373m

Answer

365m

Answer

70m

Answer

82m

Question 23

A plane is flying at an altitude of 6000m. The pilot observes the angle of depression to a landmark to be 25°. What is the horizontal distance from the plane to the landmark?

Accepted Answer

12869m

Answer

6622m

Answer

13764m

Answer

2574m

Question 24

An observer on the ground measures the angle of elevation to the top of a tower to be 32°. The observer then walks 50m towards the tower and measures the angle of elevation to be 55°. Find the height of the tower.

Accepted Answer

52m

Answer

64m

Answer

91m

Answer

39m

Question 25

A ship sails 40km due west and then 30km due south. What is the bearing of the ship from its starting point?

Accepted Answer

233°

Answer

53°

Answer

37°

Answer

323°