Applied Trigonometry: Heights, Distances & Slopes

1) A telephone pole is 18 ft tall. A brace wire runs from the top of the pole to a point 24 ft away from the base. What approximate angle does the wire make with the ground?

30°

37°

45°

53°

With rise 18 ft and run 24 ft, the angle above ground is about 37°.

Explanation

With rise 18 ft and run 24 ft, the angle above ground is about 37°.

2) A cliff is 80 m tall. A cable runs from the top to a point on the ground so that it makes a 12° angle with the ground. Approximately how far is the cable's anchor from the base?

360 m

376 m

385 m

410 m

To reach 80 m high at 12° above the ground, the anchor is about 376 m from the base.

Explanation

To reach 80 m high at 12° above the ground, the anchor is about 376 m from the base.

3) An airplane is descending at an angle of 6° toward a runway. If it is currently 2000 m above the ground, approximately how far will it travel until it reaches the runway?

19,029 m

18,600 m

21,000 m

20,300 m

From 2000 m at a 6° glide, the horizontal travel is about 19,029 m.

Explanation

From 2000 m at a 6° glide, the horizontal travel is about 19,029 m.

4) A flagpole is held by a wire 25 m long attached to its top and to the ground at a 60° angle. How tall is the flagpole?

21.7 m

22.5 m

23.4 m

20.8 m

A 25-m cable at 60° gives a vertical reach of about 21.7 m.

Explanation

A 25-m cable at 60° gives a vertical reach of about 21.7 m.

5) A person looks up to the top of a skyscraper at an angle of 75°. If the person is 100 m from the base, approximately how tall is the skyscraper?

373 m

352 m

410 m

390 m

Looking up 75° from 100 m away gives a building height near 373 m.

Explanation

Looking up 75° from 100 m away gives a building height near 373 m.

6) A cable runs from the top of a 45 ft tower to the ground at a 30° angle. How long is the cable?

50 ft

62 ft

75 ft

90 ft

A 45-ft tower with a 30° ground angle needs a cable length of about 90 ft.

Explanation

A 45-ft tower with a 30° ground angle needs a cable length of about 90 ft.

7) A mountain slope rises at an angle of 12° to the horizontal. Approximately how high does the slope rise over a horizontal distance of 500 m?

106 m

110 m

95 m

120 m

Over 500 m at a 12° grade, the rise is about 106 m.

Explanation

Over 500 m at a 12° grade, the rise is about 106 m.

8) A ski slope is 300 m long and makes a 28° angle with the ground. Approximately how high is the slope?

141 m

160 m

145 m

170 m

A 300-m slope at 28° climbs about 141 m.

Explanation

A 300-m slope at 28° climbs about 141 m.

9) A support beam is 630 m long and makes a 27° angle with the ground. How high does the beam reach?

286 m

566 m

1238 m

1440 m

Height is the vertical component of a 630-m beam at 27°; that’s about 286 m.

Explanation

Height is the vertical component of a 630-m beam at 27°; that’s about 286 m.

10) A ladder leans against a wall at an angle of 65° with the ground. The ladder is 10 m long. How high up the wall does it reach?

4.2 m

9.1 m

8.5 m

7.6 m

A 10-m ladder at 65° rises vertically by about 9.1 m.

Explanation

A 10-m ladder at 65° rises vertically by about 9.1 m.

11) A ramp is 15 ft long and makes a 37° angle with the ground. How high is the top of the ramp?

9.0 ft

8.1 ft

10.5 ft

7.3 ft

A 15-ft ramp at 37° rises by about 9.0 ft.

Explanation

A 15-ft ramp at 37° rises by about 9.0 ft.

12) A board is leaning against a wall making a 42° angle with the ground. If the board is 72 m long, how high up the wall does it reach?

34 m

42 m

48 m

50 m

A 72-m board at 42° reaches about 48 m up the wall.

Explanation

A 72-m board at 42° reaches about 48 m up the wall.

13) A kite is flying at the end of a 90 m string which makes an angle of 35° with the ground. How high is the kite above the ground?

46 m

52 m

73 m

85 m

A 90-m string at 35° lifts the kite to about 52 m.

Explanation

A 90-m string at 35° lifts the kite to about 52 m.

14) A tower is supported by a brace from the ground 50 m away. If the brace makes a 38° angle with the ground, how tall is the tower?

39 m

40 m

41 m

45 m

From 50 m out at a 38° angle, the tower’s height is about 39 m.

Explanation

From 50 m out at a 38° angle, the tower’s height is about 39 m.

15) A ramp is 12 m long and makes an angle of 20° with the ground. How high does the ramp reach?

4.1 m

5.2 m

3.2 m

6.8 m

A 12-m ramp at 20° rises by about 4.1 m.

Explanation

A 12-m ramp at 20° rises by about 4.1 m.

16) A lighthouse is 40 m tall. A support cable runs from the top to a point on the ground 15° away from vertical. How far from the base is the cable anchored?

10.4 m

10.7 m

13.8 m

14.9 m

“15° from vertical” means a small tilt; the horizontal distance is about 10.7 m from the base.

Explanation

“15° from vertical” means a small tilt; the horizontal distance is about 10.7 m from the base.

17) A 25 ft ladder makes an angle of 70° with the ground. How far is the base of the ladder from the wall?

7.5 ft

9.2 ft

8.6 ft

10.0 ft

A 25-ft ladder at 70° has its base about 8.6 ft from the wall.

Explanation

A 25-ft ladder at 70° has its base about 8.6 ft from the wall.

18) A satellite dish is angled at 42° above the horizontal. If the dish is mounted 12 ft from the ground and points to a satellite 100 ft away horizontally, what is the height above the ground where the beam hits?

98 ft

96 ft

94 ft

102 ft

The beam rises about 90 ft over the 100-ft horizontal and adds the 12-ft mount, totaling about 102 ft.

Explanation

The beam rises about 90 ft over the 100-ft horizontal and adds the 12-ft mount, totaling about 102 ft.

19) A ship's mast is 46 m tall. A rope runs from the top of the mast to the deck at a 72° angle. Approximately how far along the deck does the rope extend?

13.8 m

14.1 m

14.9 m

15.0 m

From 46 m up at 72°, the deck distance is about 15.0 m.

Explanation

From 46 m up at 72°, the deck distance is about 15.0 m.

20) A rescue plane flying at 1200 m altitude releases a supply chute at a 7° glide slope. Approximately how far horizontally will the chute travel before reaching the ground?

9,773 m

9,824 m

9,900 m

10,100 m

From 1200 m at a 7° glide, the chute travels about 9,773 m horizontally.

Explanation

From 1200 m at a 7° glide, the chute travels about 9,773 m horizontally.