Shadows & Towers Problems: Height, Shadow, and Sun Angle (HSG-SRT.C.8)

1) A 15 m flagpole casts a 20 m shadow. What is the sun's angle of elevation (nearest degree)?

37°

41°

53°

62°

tan(θ) = height / shadow = 15 / 20 = 0.75

θ = arctan(0.75) ≈ 36.87° → 37°

Hence, 37°.

Explanation

tan(θ) = height / shadow = 15 / 20 = 0.75

θ = arctan(0.75) ≈ 36.87° → 37°

Hence, 37°.

2) The sun’s elevation is 32°. A building casts a 28 m shadow. What is the building’s height (nearest meter)?

18 m

15 m

28 m

37 m

height = shadow · tan(32°) = 28 · tan(32°)

= 28 · 0.6249 ≈ 17.5 m → 18 m

Hence, 18 m.

Explanation

height = shadow · tan(32°) = 28 · tan(32°)

= 28 · 0.6249 ≈ 17.5 m → 18 m

Hence, 18 m.

3) A 6 ft sign casts a shadow of 8 ft. Which equation relates the angle of elevation θ to the data?

Tan(θ) = 6/8

Sin(θ) = 8/6

Cos(θ) = 6/8

Tan(θ) = 8/6

tan(θ) = opposite / adjacent = height / shadow = 6 / 8

Hence, tan(θ) = 6/8.

Explanation

tan(θ) = opposite / adjacent = height / shadow = 6 / 8

Hence, tan(θ) = 6/8.

4) A 10 m tree casts a shadow s when the sun’s elevation is 50°. Which expression finds s?

S = 10 tan(50°)

S = 10 sin(50°)

S = 10 cos(50°)

S = 10 / tan(50°)

tan(50°) = height / shadow = 10 / s

s = 10 / tan(50°)

Hence, s = 10 / tan(50°).

Explanation

tan(50°) = height / shadow = 10 / s

s = 10 / tan(50°)

Hence, s = 10 / tan(50°).

5) A tower is 24 m tall. If the shadow is 9 m long, what is the sun’s angle of elevation (nearest degree)?

20°

56°

69°

71°

tan(θ) = 24 / 9 = 2.666…

θ = arctan(2.666…) ≈ 69.44° → 69°

Hence, 69°.

Explanation

tan(θ) = 24 / 9 = 2.666…

θ = arctan(2.666…) ≈ 69.44° → 69°

Hence, 69°.

6) A statue casts a 3.4 m shadow when the sun’s elevation is 43°. What is the statue’s height (nearest centimeter)?

2.4 m

3.7 m

3.9 m

3.2 m

height = 3.4 · tan(43°) = 3.4 · 0.9325 ≈ 3.170 m

≈ 3.17 m → 3.2 m (nearest cm)

Hence, 3.2 m.

Explanation

height = 3.4 · tan(43°) = 3.4 · 0.9325 ≈ 3.170 m

≈ 3.17 m → 3.2 m (nearest cm)

Hence, 3.2 m.

7) The sun’s elevation is θ. A vertical pole of height H casts shadow L. Which identity is always true?

Sin(θ) = L/H

Tan(θ) = H/L

Cos(θ) = H/L

Tan(θ) = L/H

tan(θ) = opposite / adjacent = height / shadow = H / L

Hence, tan(θ) = H/L.

Explanation

tan(θ) = opposite / adjacent = height / shadow = H / L

Hence, tan(θ) = H/L.

8) A 9 m lamppost casts a shadow on level ground of length 12 m. What is the slant distance from the top to the tip of the shadow (nearest tenth)?

10.8 m

13.5 m

15.0 m

7.5 m

Slant = hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15.0

Hence, 15.0 m.

Explanation

Slant = hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15.0

Hence, 15.0 m.

9) A child 1.2 m tall casts a 2.0 m shadow at the same time a nearby tree casts a 7.5 m shadow. What is the tree’s height (nearest tenth of a meter)?

4.5 m

3.6 m

5.0 m

7.2 m

(height / shadow) is same for both.

1.2 / 2.0 = 0.6

Tree height = 0.6 · 7.5 = 4.5 m

Hence, 4.5 m.

Explanation

(height / shadow) is same for both.

1.2 / 2.0 = 0.6

Tree height = 0.6 · 7.5 = 4.5 m

Hence, 4.5 m.

10) The angle of elevation of the sun is 28°. A vertical antenna’s shadow is 16 m. What is the antenna’s height (nearest meter)?

9 m

11 m

14 m

12 m

height = 16 · tan(28°) = 16 · 0.5317 ≈ 8.51 → 9 m

Hence, 9 m.

Explanation

height = 16 · tan(28°) = 16 · 0.5317 ≈ 8.51 → 9 m

Hence, 9 m.

11) A 2.0 m sign and a 5.0 m pole stand on level ground at the same time of day. The sign’s shadow is 1.6 m. What is the pole’s shadow (nearest tenth of a meter)?

3.0 m

4.0 m

2.5 m

2.0 m

Ratio height/shadow = 2.0 / 1.6 = 1.25

Pole shadow = 5.0 / 1.25 = 4.0 m

Hence, 4.0 m.

Explanation

Ratio height/shadow = 2.0 / 1.6 = 1.25

Pole shadow = 5.0 / 1.25 = 4.0 m

Hence, 4.0 m.

12) A spotlight on the ground shines up to a wall, hitting a point 4.5 m high. If the beam forms a 35° angle with the ground, how far is the light from the wall (nearest tenth)?

6.4 m

3.7 m

7.8 m

5.7 m

tan(35°) = 4.5 / distance

distance = 4.5 / tan(35°) ≈ 4.5 / 0.7002 ≈ 6.43 → 6.4 m

Hence, 6.4 m.

Explanation

tan(35°) = 4.5 / distance

distance = 4.5 / tan(35°) ≈ 4.5 / 0.7002 ≈ 6.43 → 6.4 m

Hence, 6.4 m.

13) A 50 ft tower casts a shadow when the sun is at 40°. Which equation correctly computes the shadow length s?

S = 50 tan(40°)

S = 50 / tan(40°)

S = 50 sin(40°)

S = 50 cos(40°)

tan(40°) = 50 / s → s = 50 / tan(40°)

Hence, option B.

Explanation

tan(40°) = 50 / s → s = 50 / tan(40°)

Hence, option B.

14) At a certain time, a 1.8 m person’s shadow is 1.2 m. What is the sun’s elevation (nearest degree)?

34°

48°

52°

56°

tan(θ) = 1.8 / 1.2 = 1.5

θ = arctan(1.5) ≈ 56.31° → 56°

Hence, 56°.

Explanation

tan(θ) = 1.8 / 1.2 = 1.5

θ = arctan(1.5) ≈ 56.31° → 56°

Hence, 56°.

15) A 10 m pole casts a shadow on a ramp inclined at 12° above horizontal. If the sun’s elevation is 38°, what is the length of the shadow along the ramp (nearest tenth)?

11.8 m

13.1 m

14.9 m

10.0 m

Horizontal shadow = 10 / tan(38°) ≈ 10 / 0.7813 ≈ 12.79

Ramp length = horizontal / cos(12°) ≈ 12.79 / 0.9781 ≈ 13.08 → 13.1 m

Hence, 13.1 m.

Explanation

Horizontal shadow = 10 / tan(38°) ≈ 10 / 0.7813 ≈ 12.79

Ramp length = horizontal / cos(12°) ≈ 12.79 / 0.9781 ≈ 13.08 → 13.1 m

Hence, 13.1 m.

16) A 7 m sculpture casts a shadow of length L when the sun is at 25°. Which equation must hold?

L = 7 sin(25°)

L = 7 / tan(25°)

7 = L / tan(25°)

L = 7 cos(25°)

tan(25°) = 7 / L → L = 7 / tan(25°)

Hence, L = 7 / tan(25°).

Explanation

tan(25°) = 7 / L → L = 7 / tan(25°)

Hence, L = 7 / tan(25°).

17) The sun is at 60°. A vertical pole has a 2.0 m shadow. What is the pole’s height (nearest tenth)?

1.2 m

3.5 m

2.0 m

3.5√3 m

height = 2.0 · tan(60°) = 2.0 · √3 ≈ 3.464 → 3.5 m

Hence, 3.5 m.

Explanation

height = 2.0 · tan(60°) = 2.0 · √3 ≈ 3.464 → 3.5 m

Hence, 3.5 m.

18) A 14 ft mast casts a shadow on level ground. If the sun’s elevation increases from 30° to 45°, which statement is true?

The shadow length increases.

The slant length increases.

The mast height increases.

The shadow length decreases.

Shadow = height / tan(θ).

As θ increases, tan(θ) increases ⇒ shadow decreases.

Hence, the shadow length decreases.

Explanation

Shadow = height / tan(θ).

As θ increases, tan(θ) increases ⇒ shadow decreases.

Hence, the shadow length decreases.

19) A 20 m tower casts a 16 m shadow. What is the angle between the sun ray and the vertical (nearest degree)?

39°

51°

53°

58°

Elevation θ = arctan(20/16) ≈ arctan(1.25) ≈ 51.34°

Angle with vertical = 90° − θ ≈ 38.66° → 39°

Hence, 39°.

Explanation

Elevation θ = arctan(20/16) ≈ arctan(1.25) ≈ 51.34°

Angle with vertical = 90° − θ ≈ 38.66° → 39°

Hence, 39°.

20) A 12 m tower and a 3 m post cast shadows at the same time. The tower’s shadow is 5.4 m. What is the post’s shadow (nearest tenth of a meter)?

1.3 m

2.1 m

1.4 m

1.1 m

Ratio k = height / shadow = 12 / 5.4 = 2.222…

Post shadow = 3 / k = 3 / 2.222… ≈ 1.35 → 1.4 m

Hence, 1.4 m.

Explanation

Ratio k = height / shadow = 12 / 5.4 = 2.222…

Post shadow = 3 / k = 3 / 2.222… ≈ 1.35 → 1.4 m

Hence, 1.4 m.

Shadows and Towers: Use Trig to Find Heights & Shadow Lengths