Structural Supports and Forces

1) An engineer is designing a support beam that makes a 60° angle with the ground. If the vertical height must be 12 m, what is the beam’s length?

12 m

13.9 m

14 m

24 m

Use sin: L·sin60°=12 ⇒ L=12/0.866≈13.9 m.

Explanation

Use sin: L·sin60°=12 ⇒ L=12/0.866≈13.9 m.

2) A 10 m ladder leans against a wall, making a 75° angle with the ground. How high does it reach?

9.6 m

9.7 m

9.8 m

9.9 m

Height = 10·sin75° ≈ 9.7 m.

Explanation

Height = 10·sin75° ≈ 9.7 m.

3) A ramp rises 1.5 m over a horizontal distance of 6 m. What angle does it make with the ground (nearest degree)?

13°

14°

15°

16°

Angle = arctan(1.5/6)=arctan(0.25)≈14°.

Explanation

Angle = arctan(1.5/6)=arctan(0.25)≈14°.

4) A support cable runs from the top of a 20 m pole to the ground at a 45° angle. How long is the cable?

28 m

29 m

30 m

31 m

L=20/sin45°≈28.3 m, closest is 28 m.

Explanation

L=20/sin45°≈28.3 m, closest is 28 m.

5) A roof rafter is 8 m long, forming a 35° angle with the ceiling. How high is the roof peak above the ceiling (nearest tenth)?

4.5 m

4.6 m

4.7 m

4.8 m

Height = 8·sin35° ≈ 4.6 m.

Explanation

Height = 8·sin35° ≈ 4.6 m.

6) A guy wire is attached to the top of a 50 m tower and makes a 60° angle with the ground. How long is the wire?

57.5 m

57.6 m

57.7 m

57.8 m

L=50/sin60°≈57.7 m.

Explanation

L=50/sin60°≈57.7 m.

7) A beam 25 m long supports a platform, making a 30° angle with the ground. What vertical height does it support?

11.9 m

12.0 m

12.1 m

12.2 m

Height = 25·sin30°=12.5 m, closest is 12.2 m.

Explanation

Height = 25·sin30°=12.5 m, closest is 12.2 m.

8) A truss has a diagonal member 15 m long making a 40° angle with the horizontal. How high is it above the base?

9.6 m

9.7 m

9.8 m

9.9 m

Height = 15·sin40° ≈ 9.6 m.

Explanation

Height = 15·sin40° ≈ 9.6 m.

9) A ladder is 12 m long and placed so that it reaches 10 m high. What is its angle of elevation (nearest degree)?

56°

57°

58°

59°

sinθ=10/12 ⇒ θ≈56°.

Explanation

sinθ=10/12 ⇒ θ≈56°.

10) A cable holds a bridge deck 30 m above the river, forming a 70° angle with the deck. How long is the cable?

31.8 m

31.9 m

32.0 m

32.1 m

L=30/sin70°≈31.9 m.

Explanation

L=30/sin70°≈31.9 m.

11) A ramp makes a 20° angle with the ground. If the ramp is 5 m long, how high does it rise?

1.7 m

1.8 m

1.9 m

2.0 m

Height = 5·sin20° ≈ 1.7 m.

Explanation

Height = 5·sin20° ≈ 1.7 m.

12) A support beam 18 m long leans at a 75° angle. How high does it reach (nearest tenth)?

17.3 m

17.4 m

17.5 m

17.6 m

Height = 18·sin75° ≈ 17.4 m.

Explanation

Height = 18·sin75° ≈ 17.4 m.

13) A triangular roof has a base of 12 m and equal sloping sides of 10 m each. What is the roof’s height (nearest tenth)?

7.9 m

8.0 m

8.1 m

8.2 m

Height = √(10²−6²)=√64=8.0 m.

Explanation

Height = √(10²−6²)=√64=8.0 m.

14) A tower casts a 25 m shadow when the sun’s elevation is 65°. How tall is the tower?

53.6 m

53.7 m

53.8 m

53.9 m

Height = 25·tan65° ≈ 53.6 m.

Explanation

Height = 25·tan65° ≈ 53.6 m.

15) A wall brace makes a 45° angle with the ground and supports a wall 6 m high. How long is the brace?

8.4 m

8.5 m

8.6 m

8.7 m

L=6/sin45°≈8.5 m.

Explanation

L=6/sin45°≈8.5 m.

16) A beam is 9 m long and makes a 25° angle with the ground. What is its vertical height (nearest tenth)?

3.7 m

3.8 m

3.9 m

4.0 m

Height = 9·sin25° ≈ 3.8 m.

Explanation

Height = 9·sin25° ≈ 3.8 m.

17) A 20 m ramp rises to a platform at 12 m height. What angle with the ground does it make (nearest degree)?

36°

37°

38°

39°

sinθ=12/20=0.6 ⇒ θ≈37°.

Explanation

sinθ=12/20=0.6 ⇒ θ≈37°.

18) A guy wire supports a 15 m antenna at a 70° angle. How long is the wire (nearest tenth)?

15.9 m

16.0 m

16.1 m

16.2 m

L=15/sin70°≈16.0 m.

Explanation

L=15/sin70°≈16.0 m.

19) A scaffold forms a right triangle with base 6 m and height 8 m. How long is the diagonal support?

9.9 m

10.0 m

10.1 m

10.2 m

Hypotenuse = √(6²+8²)=10.0 m.

Explanation

Hypotenuse = √(6²+8²)=10.0 m.

20) A roof truss has a diagonal beam 22 m long that makes a 50° angle with the horizontal. What vertical height does it reach?

16.8 m

16.9 m

17.0 m

17.1 m

Height = 22·sin50° ≈ 16.9 m.

Explanation

Height = 22·sin50° ≈ 16.9 m.