Model Real Situations with Shadows & Trig � Surveying, Architecture, Navigation (HSG-SRT.C.8)

1) A 12 m lamppost casts a 7 m shadow on level ground. What is the angle of elevation of the sun (nearest degree)?

29°

40°

59°

60°

tan(θ) = 12 / 7 ≈ 1.7143

θ = arctan(1.7143) ≈ 59.74° → 60°

Hence, 60°.

Explanation

tan(θ) = 12 / 7 ≈ 1.7143

θ = arctan(1.7143) ≈ 59.74° → 60°

Hence, 60°.

2) A 30 ft tree casts an 18 ft shadow. What is the angle of elevation of the sun (nearest degree)?

48°

59°

63°

37°

tan(θ) = 30 / 18 = 1.666…

θ = arctan(1.666…) ≈ 59.04° → 59°

Hence, 59°.

Explanation

tan(θ) = 30 / 18 = 1.666…

θ = arctan(1.666…) ≈ 59.04° → 59°

Hence, 59°.

3) A building 50 m tall casts a shadow of length s when the sun’s elevation is 35°. Which equation correctly relates s and the height?

Tan(35°) = s/50

Sin(35°) = s/50

Cos(35°) = 50/s

Tan(35°) = 50/s

tan(35°) = height / shadow = 50 / s

Hence, tan(35°) = 50/s.

Explanation

tan(35°) = height / shadow = 50 / s

Hence, tan(35°) = 50/s.

4) A 1.5 m signpost casts a 2.7 m shadow. At the same time, a flagpole’s shadow is 9.0 m. How tall is the flagpole?

6.2 m

3.0 m

5.0 m

4.5 m

(height/shadow) constant ⇒ 1.5 / 2.7 = H / 9.0

H = 9.0 · (1.5/2.7) = 9.0 · (5/9) = 5.0 m

Hence, 5.0 m.

Explanation

(height/shadow) constant ⇒ 1.5 / 2.7 = H / 9.0

H = 9.0 · (1.5/2.7) = 9.0 · (5/9) = 5.0 m

Hence, 5.0 m.

5) A 20 ft tower casts a 12 ft shadow. What is the slant distance from the top of the tower to the shadow tip (nearest foot)?

20 ft

18 ft

25 ft

23 ft

Slant = √(20² + 12²) = √(400 + 144) = √544 ≈ 23.3 → 23 ft

Hence, 23 ft.

Explanation

Slant = √(20² + 12²) = √(400 + 144) = √544 ≈ 23.3 → 23 ft

Hence, 23 ft.

6) The angle of elevation from the end of a 10 m shadow to the top of a tower is 28°. What is the tower’s height (nearest meter)?

5 m

18 m

11 m

21 m

height = 10 · tan(28°) = 10 · 0.5317 ≈ 5.317 → 5 m

Hence, 5 m.

Explanation

height = 10 · tan(28°) = 10 · 0.5317 ≈ 5.317 → 5 m

Hence, 5 m.

7) From a point 40 m from the base of a building, the angle of elevation to the top is 32°. What is the building’s height above that point (nearest meter)?

25 m

21 m

34 m

63 m

height = 40 · tan(32°) = 40 · 0.6249 ≈ 25.0 m

Hence, 25 m.

Explanation

height = 40 · tan(32°) = 40 · 0.6249 ≈ 25.0 m

Hence, 25 m.

8) A 2 m survey rod has a 1.2 m shadow. At the same time, a tower’s shadow is 18 m. Estimate the tower’s height (nearest meter).

24 m

26 m

30 m

32 m

Ratio = 2 / 1.2 = 1.666…

Height = 1.666… · 18 = 30 m

Hence, 30 m.

Explanation

Ratio = 2 / 1.2 = 1.666…

Height = 1.666… · 18 = 30 m

Hence, 30 m.

9) A statue 3.5 m tall casts a 5.8 m shadow. What is tan of the sun’s elevation angle?

5.8/3.5

3.5/5.8

3.5/√(3.5²+5.8²)

√(3.5²+5.8²)/5.8

tan(θ) = opposite/adjacent = height/shadow = 3.5 / 5.8

Hence, 3.5/5.8.

Explanation

tan(θ) = opposite/adjacent = height/shadow = 3.5 / 5.8

Hence, 3.5/5.8.

10) A 48 m radio mast is on level ground. From a point where the angle of elevation to the top is 40°, how far is that point from the base (nearest meter)?

37 m

73 m

57 m

58 m

tan(40°) = 48 / x ⇒ x = 48 / tan(40°)

x ≈ 48 / 0.8391 ≈ 57.25 → 57 m

Hence, 57 m.

Explanation

tan(40°) = 48 / x ⇒ x = 48 / tan(40°)

x ≈ 48 / 0.8391 ≈ 57.25 → 57 m

Hence, 57 m.

11) The sun’s elevation is 52°. A tree’s shadow is 4.2 m. What is the tree’s height (nearest tenth of a meter)?

3.3 m

4.2 m

6.7 m

5.4 m

height = 4.2 · tan(52°) = 4.2 · 1.279 ≈ 5.372 → 5.4 m

Hence, 5.4 m.

Explanation

height = 4.2 · tan(52°) = 4.2 · 1.279 ≈ 5.372 → 5.4 m

Hence, 5.4 m.

12) A tower stands on a hill sloped at 8° above horizontal. From a point downhill, the angle of elevation to the tower’s base is 8° and to the top is 25°. The horizontal distance to the base is 60 m. What is the tower’s height (nearest meter)?

20 m

26 m

22 m

29 m

Vertical to base = 60 · tan(8°)

Vertical to top = 60 · tan(25°)

Height = 60( tan25° − tan8° ) ≈ 60(0.4663 − 0.1405) ≈ 19.55 → 20 m

Hence, 20 m.

Explanation

Vertical to base = 60 · tan(8°)

Vertical to top = 60 · tan(25°)

Height = 60( tan25° − tan8° ) ≈ 60(0.4663 − 0.1405) ≈ 19.55 → 20 m

Hence, 20 m.

13) A 6 ft person stands 10 ft from a lamp post. The tip of the person’s shadow meets the tip of the lamp post’s shadow at the same point 22 ft from the post. How tall is the lamp post (nearest foot)?

8 ft

11 ft

20 ft

24 ft

Person shadow = 22 − 10 = 12 ft

Similar triangles: 6 / H = 12 / 22

H = 6 · (22 / 12) = 11 ft

Hence, 11 ft.

Explanation

Person shadow = 22 − 10 = 12 ft

Similar triangles: 6 / H = 12 / 22

H = 6 · (22 / 12) = 11 ft

Hence, 11 ft.

14) The sun’s elevation increases from 30° to 45°. For a 10 m pole, s1 is the shadow length at 30° and s2 at 45°. Which is true?

S1 = s2

S1 < s2

S1 = 10 s2

S1 > s2

s = h / tan(θ).

an(30°) s2.

Hence, s1 > s2.

Explanation

s = h / tan(θ).

an(30°) s2.

Hence, s1 > s2.

15) A building 80 m tall casts a slanted ray from its top to the shadow tip of length 120 m. What is sin of the sun’s elevation angle?

120/80

80/120

80/√(80^2+120^2)

√(80^2+120^2)/120

sin(θ) = opposite / hypotenuse = 80 / 120 = 2/3

Hence, 80/120.

Explanation

sin(θ) = opposite / hypotenuse = 80 / 120 = 2/3

Hence, 80/120.

16) A 25 ft pole on level ground casts a 7 ft shadow. What is the sun’s elevation (nearest degree)?

70°

74°

68°

76°

tan(θ) = 25 / 7 ≈ 3.5714

θ = arctan(3.5714) ≈ 74.05° → 74°

Hence, 74°.

Explanation

tan(θ) = 25 / 7 ≈ 3.5714

θ = arctan(3.5714) ≈ 74.05° → 74°

Hence, 74°.

17) From a point 24 m from a tower’s base, the angle of elevation to the top is 53°. What is the tower’s height (nearest meter)?

18 m

24 m

30 m

32 m

height = 24 · tan(53°) = 24 · 1.327 ≈ 31.85 → 32 m

Hence, 32 m.

Explanation

height = 24 · tan(53°) = 24 · 1.327 ≈ 31.85 → 32 m

Hence, 32 m.

18) A 1.8 m person views the top of a monument at an elevation angle of 35° from a point 28 m from the base. What is the total height of the monument (nearest meter)?

19 m

23 m

21 m

25 m

Rise above eye = 28 · tan(35°) ≈ 28 · 0.7002 ≈ 19.6 m

Total height = 1.8 + 19.6 ≈ 21.4 → 21 m

Hence, 21 m.

Explanation

Rise above eye = 28 · tan(35°) ≈ 28 · 0.7002 ≈ 19.6 m

Total height = 1.8 + 19.6 ≈ 21.4 → 21 m

Hence, 21 m.

19) At the same time of day, a pole’s shadow is 3.2 m and a building’s shadow is 24.0 m. If the pole is 4.0 m tall, how tall is the building (nearest tenth of a meter)?

24.0 m

25.0 m

30.0 m

33.0 m

Ratio = height/shadow = 4.0 / 3.2 = 1.25

Building height = 1.25 · 24.0 = 30.0 m

Hence, 30.0 m.

Explanation

Ratio = height/shadow = 4.0 / 3.2 = 1.25

Building height = 1.25 · 24.0 = 30.0 m

Hence, 30.0 m.

20) A cliff casts an 85 m shadow. From the shadow tip, the elevation angle to the top of the cliff is 27°. What is the cliff’s height (nearest meter)?

43 m

38 m

45 m

50 m

height = 85 · tan(27°) = 85 · 0.5095 ≈ 43.3 → 43 m

Hence, 43 m.

Explanation

height = 85 · tan(27°) = 85 · 0.5095 ≈ 43.3 → 43 m

Hence, 43 m.