Compare Shadows to Find Unknown Heights: Proportional Reasoning & Trig (HSG-SRT.C.8)

1) A mural wall of unknown height H casts a shadow S. If at the same time a 2.0 m stick casts a 0.8 m shadow, which equation relates H and S?

H/S = 0.8/2.0

H/S = 2.0/0.8

H S = 2.0 + 0.8

H/S = sin(θ)

H/S = height/shadow (same angle) = 2.0 / 0.8 = 2.5

Hence, H/S = 2.0/0.8.

Explanation

H/S = height/shadow (same angle) = 2.0 / 0.8 = 2.5

Hence, H/S = 2.0/0.8.

2) A 3.2 m pole casts a 1.0 m shadow at noon. Later, the same pole casts a 2.0 m shadow. Which statement is true about the later sun angle?

Increased (sun higher)

Decreased (sun lower)

Same

Became 90°

Longer shadow ⇒ smaller elevation angle.

Hence, decreased (sun lower).

Explanation

Longer shadow ⇒ smaller elevation angle.

Hence, decreased (sun lower).

3) The sun’s elevation is 50°. A 6 m flagpole casts a shadow L1. Which is the shadow L2 of a 9 m flagpole at the same time?

L2 = L1

L2 = 1.5 L1

L2 = 1.25 L1

L2 = 0.75 L1

Shadows ∝ heights (same angle).

9 m is 1.5 × 6 m ⇒ L2 = 1.5 L1

Hence, 1.5 L1.

Explanation

Shadows ∝ heights (same angle).

9 m is 1.5 × 6 m ⇒ L2 = 1.5 L1

Hence, 1.5 L1.

4) A lamppost 4.5 m tall casts a 3.0 m shadow. Another object has height 7.5 m. What is its shadow (nearest tenth of a meter)?

4.7 m

5.0 m

6.0 m

3.0 m

Ratio = 4.5 / 3.0 = 1.5

Shadow = 7.5 / 1.5 = 5.0 m

Hence, 5.0 m.

Explanation

Ratio = 4.5 / 3.0 = 1.5

Shadow = 7.5 / 1.5 = 5.0 m

Hence, 5.0 m.

5) Two trees, A and B, stand on level ground. A is 8 m tall with a 5 m shadow. B has a 12.5 m shadow. What is B’s height?

18.0 m

19.5 m

20.0 m

17.0 m

Ratio = 8 / 5 = 1.6

Height_B = 1.6 · 12.5 = 20.0 m

Hence, 20.0 m.

Explanation

Ratio = 8 / 5 = 1.6

Height_B = 1.6 · 12.5 = 20.0 m

Hence, 20.0 m.

6) A 1.5 m post casts a shadow of length s1 at sun angle θ. A 4.5 m pole casts a shadow s2 at the same time. Which relation is correct?

S2 = s1

S2 = 3 s1

S2 = s1/3

S2 = s1 tan(θ)

Shadows ∝ heights ⇒ s2 / s1 = 4.5 / 1.5 = 3

Hence, s2 = 3 s1.

Explanation

Shadows ∝ heights ⇒ s2 / s1 = 4.5 / 1.5 = 3

Hence, s2 = 3 s1.

7) A 2.0 m bollard casts a 1.2 m shadow. A sculpture is 3.5 m tall. What is the sculpture’s shadow length (nearest tenth of a meter)?

1.7 m

2.1 m

2.4 m

2.9 m

Ratio = 2.0 / 1.2 = 1.666…

Shadow = 3.5 / 1.666… = 3.5 · (3/5) = 2.1 m

Nearest tenth: 2.1 m → Option B

Hence, 2.1 m.

Explanation

Ratio = 2.0 / 1.2 = 1.666…

Shadow = 3.5 / 1.666… = 3.5 · (3/5) = 2.1 m

Nearest tenth: 2.1 m → Option B

Hence, 2.1 m.

8) On a certain day, tan(θ) = 0.8. An object’s height is H and its shadow is L. Which statement is true?

L = 0.8 H

H = 0.8 L

H/L = 0.8

L/H = 0.8

tan(θ) = height / shadow = H / L = 0.8

Hence, H/L = 0.8.

Explanation

tan(θ) = height / shadow = H / L = 0.8

Hence, H/L = 0.8.

9) A 12 ft post casts a shadow of 9 ft. At the same time, another post has height 18 ft. What is its shadow (nearest tenth of a foot)?

11.5 ft

12.0 ft

13.5 ft

14.0 ft

Ratio = 12 / 9 = 4/3

Shadow = 18 / (4/3) = 18 · (3/4) = 13.5 ft

Hence, 13.5 ft.

Explanation

Ratio = 12 / 9 = 4/3

Shadow = 18 / (4/3) = 18 · (3/4) = 13.5 ft

Hence, 13.5 ft.

10) Two vertical poles cast shadows on level ground at the same time. Pole A: height 7 m, shadow 4 m. Pole B: height 10.5 m, shadow x. Find x.

5.0 m

5.5 m

6.0 m

6.5 m

x / 4 = 10.5 / 7 = 1.5

x = 4 · 1.5 = 6.0 m

Hence, 6.0 m.

Explanation

x / 4 = 10.5 / 7 = 1.5

x = 4 · 1.5 = 6.0 m

Hence, 6.0 m.

11) The sun’s elevation is θ. Object X has height 5 m and shadow 3 m. Which equation expresses tan(θ)?

Tan(θ) = 3/5

Tan(θ) = 5/3

Tan(θ) = 5

Tan(θ) = 3

tan(θ) = height / shadow = 5 / 3

Hence, 5/3.

Explanation

tan(θ) = height / shadow = 5 / 3

Hence, 5/3.

12) A 20 m mast casts a 12 m shadow. Later, the shadow is 15 m. What happened to the sun’s elevation?

It increased.

It decreased.

It stayed the same.

It became 90°.

Longer shadow ⇒ smaller elevation angle.

Hence, it decreased.

Explanation

Longer shadow ⇒ smaller elevation angle.

Hence, it decreased.

13) A 2.5 m sign casts a 1.0 m shadow. A nearby tower casts a 16.0 m shadow. What is the tower’s height?

35 m

37.5 m

40 m

32 m

Ratio = height/shadow = 2.5 / 1.0 = 2.5

Tower height = 2.5 · 16.0 = 40 m

Hence, 40 m.

Explanation

Ratio = height/shadow = 2.5 / 1.0 = 2.5

Tower height = 2.5 · 16.0 = 40 m

Hence, 40 m.

14) When the sun’s elevation is 30°, a 9 m tower’s shadow is L. When the elevation becomes 60°, what is the new shadow length in terms of L?

L

L/√3

L/3

√3 L

L₁ = 9 / tan30° = 9 / (√3/3) = 9√3

L₂ = 9 / tan60° = 9 / √3 = 3√3

L₂ / L₁ = (3√3) / (9√3) = 1/3 ⇒ L₂ = L/3

Hence, L/3.

Explanation

L₁ = 9 / tan30° = 9 / (√3/3) = 9√3

L₂ = 9 / tan60° = 9 / √3 = 3√3

L₂ / L₁ = (3√3) / (9√3) = 1/3 ⇒ L₂ = L/3

Hence, L/3.

15) A 10 ft post casts a 7 ft shadow. At the same time, a statue casts a 9.8 ft shadow. What is the statue’s height (nearest tenth of a foot)?

14.0 ft

13.0 ft

14.5 ft

15.0 ft

(height / shadow) constant ⇒ 10 / 7 = x / 9.8

x = 9.8 · (10/7) = 14.0 ft

Hence, 14.0 ft.

Explanation

(height / shadow) constant ⇒ 10 / 7 = x / 9.8

x = 9.8 · (10/7) = 14.0 ft

Hence, 14.0 ft.

16) The sun’s elevation is θ. Object A has height 3 m and shadow 2 m. Which expression gives the shadow of Object B with height 4.5 m?

3.5

4.5

2

3

tan(θ) = 3 / 2 ⇒ shadow = height / tan(θ)

Shadow_B = 4.5 / (3/2) = 4.5 · (2/3) = 3

Hence, 3.

Explanation

tan(θ) = 3 / 2 ⇒ shadow = height / tan(θ)

Shadow_B = 4.5 / (3/2) = 4.5 · (2/3) = 3

Hence, 3.

17) Two towers stand on level ground when the sun’s elevation is 35°. Tower T1 is 24 m tall. Which equation gives T1’s shadow length s1?

S1 = 24 tan(35°)

S1 = 24 sin(35°)

S1 = 24 / tan(35°)

S1 = 24 / cos(35°)

tan(35°) = 24 / s1 ⇒ s1 = 24 / tan(35°)

Hence, option C.

Explanation

tan(35°) = 24 / s1 ⇒ s1 = 24 / tan(35°)

Hence, option C.

18) A sign 2.4 m tall casts a 1.5 m shadow. A nearby building casts a 17.5 m shadow at the same time. What is the building’s height?

26.0 m

27.5 m

28.0 m

28.5 m

Ratio = 2.4 / 1.5 = 1.6

Building height = 1.6 · 17.5 = 28.0 m

Hence, 28.0 m.

Explanation

Ratio = 2.4 / 1.5 = 1.6

Building height = 1.6 · 17.5 = 28.0 m

Hence, 28.0 m.

19) On level ground, Object A: height 5 m, shadow 4 m. Object B has shadow 10 m at the same time. What is Object B’s height?

11.5 m

12.0 m

12.5 m

10.0 m

Ratio = 5 / 4 = 1.25

Height_B = 1.25 · 10 = 12.5 m

Hence, 12.5 m.

Explanation

Ratio = 5 / 4 = 1.25

Height_B = 1.25 · 10 = 12.5 m

Hence, 12.5 m.

20) A 1.6 m pole casts a 2.0 m shadow. At the same time, a tree's shadow is 7.5 m. What is the tree's height (nearest tenth of a meter)?

5.6 m

5.8 m

6.0 m

6.2 m

(height / shadow) constant ⇒ 1.6 / 2.0 = 0.8

Tree height = 0.8 · 7.5 = 6.0 m

Hence, 6.0 m.

Explanation

(height / shadow) constant ⇒ 1.6 / 2.0 = 0.8

Tree height = 0.8 · 7.5 = 6.0 m

Hence, 6.0 m.

Similar Triangles & Proportions with Shadows

1) A mural wall of unknown height H casts a shadow S. If at the same time a 2.0 m stick casts a 0.8 m shadow, which equation relates H and S?

2)

What first name or nickname would you like us to use?

2) A 3.2 m pole casts a 1.0 m shadow at noon. Later, the same pole casts a 2.0 m shadow. Which statement is true about the later sun angle?

3) The sun’s elevation is 50°. A 6 m flagpole casts a shadow L1. Which is the shadow L2 of a 9 m flagpole at the same time?

4) A lamppost 4.5 m tall casts a 3.0 m shadow. Another object has height 7.5 m. What is its shadow (nearest tenth of a meter)?

5) Two trees, A and B, stand on level ground. A is 8 m tall with a 5 m shadow. B has a 12.5 m shadow. What is B’s height?

6) A 1.5 m post casts a shadow of length s1 at sun angle θ. A 4.5 m pole casts a shadow s2 at the same time. Which relation is correct?

7) A 2.0 m bollard casts a 1.2 m shadow. A sculpture is 3.5 m tall. What is the sculpture’s shadow length (nearest tenth of a meter)?

8) On a certain day, tan(θ) = 0.8. An object’s height is H and its shadow is L. Which statement is true?

9) A 12 ft post casts a shadow of 9 ft. At the same time, another post has height 18 ft. What is its shadow (nearest tenth of a foot)?

10) Two vertical poles cast shadows on level ground at the same time. Pole A: height 7 m, shadow 4 m. Pole B: height 10.5 m, shadow x. Find x.

11) The sun’s elevation is θ. Object X has height 5 m and shadow 3 m. Which equation expresses tan(θ)?

12) A 20 m mast casts a 12 m shadow. Later, the shadow is 15 m. What happened to the sun’s elevation?

13) A 2.5 m sign casts a 1.0 m shadow. A nearby tower casts a 16.0 m shadow. What is the tower’s height?

14) When the sun’s elevation is 30°, a 9 m tower’s shadow is L. When the elevation becomes 60°, what is the new shadow length in terms of L?

15) A 10 ft post casts a 7 ft shadow. At the same time, a statue casts a 9.8 ft shadow. What is the statue’s height (nearest tenth of a foot)?

16) The sun’s elevation is θ. Object A has height 3 m and shadow 2 m. Which expression gives the shadow of Object B with height 4.5 m?

17) Two towers stand on level ground when the sun’s elevation is 35°. Tower T1 is 24 m tall. Which equation gives T1’s shadow length s1?

18) A sign 2.4 m tall casts a 1.5 m shadow. A nearby building casts a 17.5 m shadow at the same time. What is the building’s height?

19) On level ground, Object A: height 5 m, shadow 4 m. Object B has shadow 10 m at the same time. What is Object B’s height?

20) A 1.6 m pole casts a 2.0 m shadow. At the same time, a tree's shadow is 7.5 m. What is the tree's height (nearest tenth of a meter)?