Real-World Right-Triangle Problems � Heights & Distances (HSF-TF.C.8)

1) A ladder leans against a wall making a 70° angle with the ground. The ladder is 10 ft long. How high does it reach?

9.5 ft

10 ft

9.4 ft

8.9 ft

height = 10 × sin 70° ≈ 10 × 0.9397 = 9.4. Hence, height ≈ 9.4 ft.

Explanation

height = 10 × sin 70° ≈ 10 × 0.9397 = 9.4. Hence, height ≈ 9.4 ft.

2) A ramp 15 ft long rises at a 25° angle. Find the vertical height of the ramp.

5.8 ft

6.0 ft

7.1 ft

6.3 ft

height = 15 × sin 25° ≈ 15 × 0.4226 = 6.3. Hence, height ≈ 6.3 ft.

Explanation

height = 15 × sin 25° ≈ 15 × 0.4226 = 6.3. Hence, height ≈ 6.3 ft.

3) A radio tower 80 m tall casts a shadow of 55 m. Find the angle of elevation of the sun to the nearest degree.

43°

56°

49°

61°

tan θ = 80 / 55 = 1.4545; θ = arctan(1.4545) ≈ 56°.

Explanation

tan θ = 80 / 55 = 1.4545; θ = arctan(1.4545) ≈ 56°.

4) A surveyor measures a hill that rises 30 m vertically over a horizontal distance of 200 m. What is the hill’s angle of elevation?

7°

8°

9°

11°

tan θ = 30 / 200 = 0.15; θ = arctan(0.15) ≈ 9°.

Explanation

tan θ = 30 / 200 = 0.15; θ = arctan(0.15) ≈ 9°.

5) A 20 m guy wire is attached to a pole and makes a 60° angle with the ground. How tall is the pole?

15 m

17.3 m

18 m

height = 20 × sin 60° = 20 × (√3/2) = 17.3 m.

Explanation

height = 20 × sin 60° = 20 × (√3/2) = 17.3 m.

6) A bridge rises 4 m for every 40 m of horizontal distance. What is the angle of elevation of the bridge surface?

4°

5°

5.7°

6°

tan θ = 4 / 40 = 0.1; θ = arctan(0.1) ≈ 5.7°.

Explanation

tan θ = 4 / 40 = 0.1; θ = arctan(0.1) ≈ 5.7°.

7) A kite is flying 50 m above the ground. The string makes a 35° angle with the ground. How long is the string?

60.5 m

85.7 m

88.3 m

87.1 m

length = 50 / sin 35° = 87.1 m.

Explanation

length = 50 / sin 35° = 87.1 m.

8) A building casts a 16 m shadow. The sun’s angle of elevation is 65°. Find the height of the building.

30 m

33.4 m

34.4 m

32.2 m

height = 16 × tan 65° = 34.4 m.

Explanation

height = 16 × tan 65° = 34.4 m.

9) A tree casts a 22 m shadow when the sun’s angle of elevation is 40°. Find the height of the tree.

17.0 m

18.5 m

20.8 m

15.8 m

height = 22 × tan 40° = 18.5 m.

Explanation

height = 22 × tan 40° = 18.5 m.

10) A 12 ft ramp rises 2.5 ft. Find the ramp’s angle of elevation.

10°

12°

14°

13°

sin θ = 2.5 / 12 = 0.2083; θ = arcsin(0.2083) ≈ 12°.

Explanation

sin θ = 2.5 / 12 = 0.2083; θ = arcsin(0.2083) ≈ 12°.

11) A ship is anchored 200 m from a lighthouse. The top of the lighthouse is seen at a 20° angle of elevation. How tall is the lighthouse?

65.2 m

72.8 m

70.5 m

60.9 m

height = 200 × tan 20° = 72.8 m.

Explanation

height = 200 × tan 20° = 72.8 m.

12) A 30 ft telephone pole leans so that its top is 28 ft above the ground. Find the angle the pole makes with the ground.

20°

69.7°

24°

26°

sin θ = 28 / 30 = 0.9333; To find the angle take the inverse sine (arcsin) of both sides of the equation. θ = arcsin(28/30) ≈ 69.7°, not listed.

Explanation

sin θ = 28 / 30 = 0.9333; To find the angle take the inverse sine (arcsin) of both sides of the equation. θ = arcsin(28/30) ≈ 69.7°, not listed.

13) A ski slope drops 120 m vertically over a slope distance of 400 m. What is the angle of descent?

15°

16°

17°

18°

sin θ = 120 / 400 = 0.3; θ = arcsin(0.3) ≈ 17°.

Explanation

sin θ = 120 / 400 = 0.3; θ = arcsin(0.3) ≈ 17°.

14) A 40 ft ladder reaches a window 30 ft above the ground. What angle does the ladder make with the ground?

40°

49°

51°

60°

sin θ = 30 / 40 = 0.75; θ = arcsin(0.75) ≈ 49°.

Explanation

sin θ = 30 / 40 = 0.75; θ = arcsin(0.75) ≈ 49°.

15) A building 50 m tall casts a shadow 60 m long. What is the angle of elevation of the sun?

40°

39°

41°

45°

tan θ = 50 / 60 = 0.8333; θ = arctan(0.8333) ≈ 40°.

Explanation

tan θ = 50 / 60 = 0.8333; θ = arctan(0.8333) ≈ 40°.

16) A rescue cable extends from a helicopter at a 30° angle to the ground. If the helicopter is hovering 100 ft above the ground, how long is the cable?

180 ft

200 ft

150 ft

175 ft

length = 100 / sin 30° = 200 ft.

Explanation

length = 100 / sin 30° = 200 ft.

17) A 25 m ramp has an angle of elevation of 10°. What is its vertical rise?

4.2 m

4.3 m

5.0 m

3.9 m

rise = 25 × sin 10° ≈ 4.3 m.

Explanation

rise = 25 × sin 10° ≈ 4.3 m.

18) A cliff rises vertically 150 m above the sea. From a boat, the angle of elevation to the top is 25°. How far is the boat from the base of the cliff?

300 m

321 m

280 m

330 m

distance = 150 / tan 25° = 321 m.

Explanation

distance = 150 / tan 25° = 321 m.