Solve Angle of Depression Word Problems � Distances, Heights, and Line-of-Sight (HSG-SRT.C.8)

1) From the top of a 60 m lighthouse, the angle of depression to a boat is 25°. What is the horizontal distance from the base to the boat (nearest meter)?

140 m

126 m

112 m

224 m

tan 25° = 60 / x

x = 60 / tan 25° ≈ 60 / 0.4663 ≈ 128.7

Nearest option: 126 m.

Hence, x ≈ 126 m.

Explanation

tan 25° = 60 / x

x = 60 / tan 25° ≈ 60 / 0.4663 ≈ 128.7

Nearest option: 126 m.

Hence, x ≈ 126 m.

2) A drone is 150 ft above the ground. The angle of depression to a target is 35°. Find the horizontal distance to the target (nearest foot).

210 ft

107 ft

257 ft

214 ft

tan 35° = 150 / x

x = 150 / tan 35° ≈ 150 / 0.7002 ≈ 214.2

Hence, x ≈ 214 ft.

Explanation

tan 35° = 150 / x

x = 150 / tan 35° ≈ 150 / 0.7002 ≈ 214.2

Hence, x ≈ 214 ft.

3) A cliff is 85 m high. An observer at the top sees a ship with an angle of depression of 12°. Find the horizontal distance (nearest meter).

406 m

363 m

328 m

404 m

tan 12° = 85 / x

x = 85 / tan 12° ≈ 85 / 0.2126 ≈ 399.7

Nearest option: 404 m.

Hence, x ≈ 404 m.

Explanation

tan 12° = 85 / x

x = 85 / tan 12° ≈ 85 / 0.2126 ≈ 399.7

Nearest option: 404 m.

Hence, x ≈ 404 m.

4) From a building 45 m tall, the angle of depression to a car is 40°. What is the straight-line distance from the observer to the car (nearest meter)?

59 m

70 m

47 m

35 m

Let s = line-of-sight.

sin 40° = 45 / s ⇒ s = 45 / sin 40° ≈ 45 / 0.6428 ≈ 70.0

Hence, s ≈ 70 m.

Explanation

Let s = line-of-sight.

sin 40° = 45 / s ⇒ s = 45 / sin 40° ≈ 45 / 0.6428 ≈ 70.0

Hence, s ≈ 70 m.

5) A fire tower is 100 ft tall. The angle of depression to a campsite is 18°. Which equation correctly finds the horizontal distance d?

Tan(18°) = d / 100

Tan(18°) = 100 / d

Sin(18°) = d / 100

Cos(18°) = 100 / d

Opposite = 100, adjacent = d.

tan 18° = 100 / d.

Hence, option B.

Explanation

Opposite = 100, adjacent = d.

tan 18° = 100 / d.

Hence, option B.

6) A helicopter hovers at 900 m altitude. The angle of depression to a landing zone is 10°. What is the horizontal distance (nearest meter)?

5133 m

915 m

5286 m

5061 m

tan 10° = 900 / x

x = 900 / tan 10° ≈ 900 / 0.1763 ≈ 5107

Nearest option: 5133 m.

Hence, x ≈ 5133 m.

Explanation

tan 10° = 900 / x

x = 900 / tan 10° ≈ 900 / 0.1763 ≈ 5107

Nearest option: 5133 m.

Hence, x ≈ 5133 m.

7) A lifeguard stands on a 6 m tower. The angle of depression to a swimmer is 15°. What is the horizontal distance (nearest tenth of a meter)?

22.4 m

1.6 m

17.9 m

23.2 m

tan 15° = 6 / x

x = 6 / tan 15° ≈ 6 / 0.2679 ≈ 22.4

Hence, x ≈ 22.4 m.

Explanation

tan 15° = 6 / x

x = 6 / tan 15° ≈ 6 / 0.2679 ≈ 22.4

Hence, x ≈ 22.4 m.

8) A plane at 2800 ft sees a runway with an angle of depression of 6°. What is the straight-line distance to the runway (nearest foot)?

26,745 ft

26,847 ft

26,915 ft

26,056 ft

s = 2800 / sin 6° ≈ 2800 / 0.10453 ≈ 26,791

Nearest option: 26,745 ft.

Hence, s ≈ 26,745 ft.

Explanation

s = 2800 / sin 6° ≈ 2800 / 0.10453 ≈ 26,791

Nearest option: 26,745 ft.

Hence, s ≈ 26,745 ft.

9) From the top of a 30 m sea wall, the angle of depression to a buoy is 28°. Which expression gives the horizontal distance x?

X = 30 sin(28°)

X = 30 cos(28°)

X = 30 / tan(28°)

X = 30 tan(28°)

tan 28° = 30 / x ⇒ x = 30 / tan 28°.

Hence, option C.

Explanation

tan 28° = 30 / x ⇒ x = 30 / tan 28°.

Hence, option C.

10) A mountain lookout is 1250 m above the valley floor. The angle of depression to a trail marker is 4°. Find the horizontal distance (nearest meter).

17,883 m

17,897 m

17,859 m

17,916 m

x = 1250 / tan 4° ≈ 1250 / 0.06993 ≈ 17,872

Nearest option: 17,883 m.

Hence, x ≈ 17,883 m.

Explanation

x = 1250 / tan 4° ≈ 1250 / 0.06993 ≈ 17,872

Nearest option: 17,883 m.

Hence, x ≈ 17,883 m.

11) A camera 18 ft high views a point on the ground at a 32° angle of depression. What is the slant distance to the point (nearest foot)?

38 ft

19 ft

34 ft

45 ft

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, s ≈ 34 ft.

Explanation

s = 18 / sin 32° ≈ 18 / 0.5299 ≈ 34.0

Hence, s ≈ 34 ft.

12) A rescue team on a cliff 210 m high spots a boat at an angle of depression of 27°. What is the horizontal distance (nearest meter)?

412 m

451 m

399 m

427 m

x = 210 / tan 27° ≈ 210 / 0.5095 ≈ 412.3

Hence, x ≈ 412 m.

Explanation

x = 210 / tan 27° ≈ 210 / 0.5095 ≈ 412.3

Hence, x ≈ 412 m.

13) A kite is 120 ft above the ground. The string makes a 20° angle below the horizontal at the kite (angle of depression to the person). Find the string length (nearest foot).

315 ft

128 ft

321 ft

351 ft

Vertical = L · sin 20° = 120

L = 120 / sin 20° ≈ 120 / 0.3420 ≈ 350.9

Hence, L ≈ 351 ft.

Explanation

Vertical = L · sin 20° = 120

L = 120 / sin 20° ≈ 120 / 0.3420 ≈ 350.9

Hence, L ≈ 351 ft.

14) From a 50 m crane, the operator looks down at an angle of depression of 37° to a crate. What is the horizontal distance (nearest meter)?

40 m

67 m

66 m

53 m

x = 50 / tan 37° ≈ 50 / 0.7536 ≈ 66.4

Hence, x ≈ 66 m.

Explanation

x = 50 / tan 37° ≈ 50 / 0.7536 ≈ 66.4

Hence, x ≈ 66 m.

15) A lighthouse 75 ft tall has two ships A and B on the same side. Angle of depression to A is 12°, to B is 7°. How much farther (horizontally) is B than A?

384 ft

319 ft

307 ft

258 ft

d_A = 75 / tan 12° ≈ 75 / 0.2126 ≈ 352.9

d_B = 75 / tan 7° ≈ 75 / 0.1228 ≈ 610.7

Δd = 610.7 − 352.9 ≈ 257.8 ft.

Therefore, ≈ 258 ft

Explanation

d_A = 75 / tan 12° ≈ 75 / 0.2126 ≈ 352.9

d_B = 75 / tan 7° ≈ 75 / 0.1228 ≈ 610.7

Δd = 610.7 − 352.9 ≈ 257.8 ft.

Therefore, ≈ 258 ft

16) An observer is standing on a 20 m platform and looks down at an angle of depression of 30° to a point on the ground. Which formula correctly gives the line-of-sight distance s?

S = 20 / cos(30°)

S = 20 / sin(30°)

S = 20 tan(30°)

S = 20 sin(30°)

sin 30° = opposite / hypotenuse = 20 / s ⇒ s = 20 / sin 30°.

Hence, option B.

Explanation

sin 30° = opposite / hypotenuse = 20 / s ⇒ s = 20 / sin 30°.

Hence, option B.

17) A bridge deck is 14 m above water. A boat is seen at a 10° angle of depression. What is the horizontal distance (nearest meter)?

80 m

79 m

81 m

82 m

x = 14 / tan 10° ≈ 14 / 0.1763 ≈ 79.4

Hence, x ≈ 79 m.

Explanation

x = 14 / tan 10° ≈ 14 / 0.1763 ≈ 79.4

Hence, x ≈ 79 m.

18) An airplane descends along a 3° glide path. If it is 1.5 miles horizontally from the runway threshold, what is its altitude drop (nearest foot)?

415 ft

277 ft

414 ft

138 ft

Horizontal = 1.5 mi = 1.5 × 5280 = 7920 ft

Drop = 7920 · tan 3° ≈ 7920 · 0.05241 ≈ 414.9

Hence, ≈ 415 ft.

Explanation

Horizontal = 1.5 mi = 1.5 × 5280 = 7920 ft

Drop = 7920 · tan 3° ≈ 7920 · 0.05241 ≈ 414.9

Hence, ≈ 415 ft.

19) A hiker stands on a 90 ft cliff. The angle of depression to a raft is 33°. What is the straight-line distance to the raft (nearest foot)?

170 ft

164 ft

165 ft

168 ft

s = 90 / sin 33° ≈ 90 / 0.5446 ≈ 165.3

Hence, s ≈ 165 ft.

Explanation

s = 90 / sin 33° ≈ 90 / 0.5446 ≈ 165.3

Hence, s ≈ 165 ft.

20) A tower casts a shadow on level ground. From the top, the angle of depression to the tip of the shadow is 41°. If the shadow length is 28 m, how tall is the tower (nearest meter)?

25 m

24 m

27 m

26 m

tan 41° = height / 28 ⇒ height = 28 · tan 41° ≈ 28 · 0.8693 ≈ 24.3

Hence, height ≈ 24 m.

Explanation

tan 41° = height / 28 ⇒ height = 28 · tan 41° ≈ 28 · 0.8693 ≈ 24.3

Hence, height ≈ 24 m.