Height From Shadow Quiz: Height From Shadow Using Ratio

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2) If both shadows double in length while heights stay the same, the ratio method still yields the same heights.

True

False

The ratio shadow₂/shadow₁ is unchanged, so computed heights are consistent.

Explanation

The ratio shadow₂/shadow₁ is unchanged, so computed heights are consistent.

3) If two vertical objects stand on level ground at the same time of day, their height-to-shadow ratios are equal.

True

False

Same sun angle θ gives tanθ = height/shadow for both, so ratios are equal.

Explanation

Same sun angle θ gives tanθ = height/shadow for both, so ratios are equal.

4) When does the shadow-ratio method fail or need adjustment? Select all that apply.

Objects are not vertical (leaning)

Ground is not level

Measurements are taken at different times with changing sun angle

Shadows are obstructed or not fully measurable

Heights include slanted structures instead of vertical height

The method relies on vertical objects, level ground, the same sun angle, clear shadows, and true vertical height.

Explanation

The method relies on vertical objects, level ground, the same sun angle, clear shadows, and true vertical height.

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6) Which are common mistakes to avoid when using height₁/shadow₁ = height₂/shadow₂?

Swapping height and shadow in the ratio

Using different length units for the two shadows

Measuring at different times with different sun angles

Using slant object length instead of vertical height

Rounding too early when computing ratios

All listed issues can invalidate or degrade the proportional calculation.

Explanation

All listed issues can invalidate or degrade the proportional calculation.

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7) Which data sets are sufficient to compute an unknown height using the ratio method?

(height₁, shadow₁, shadow₂)

(height₁, height₂, shadow₁)

(height₁, shadow₂)

(shadow₁, shadow₂)

(height₂ only)

Either give three of the four variables in the proportion to solve for the fourth.

Explanation

Either give three of the four variables in the proportion to solve for the fourth.

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8) If height₁/shadow₁ = height₂/shadow₂, then shadow₂ = shadow₁·(height₂/height₁).

True

False

Rearrange the proportion to solve for shadow₂.

Explanation

Rearrange the proportion to solve for shadow₂.

9) A 3.2 m sign casts a 2.0 m shadow. A tower’s shadow is 5.0 m at the same time. Find the tower height.

7.6 m

8.0 m

8.4 m

9.0 m

height₂ = 3.2·(5.0/2.0) = 8.0 m.

Explanation

height₂ = 3.2·(5.0/2.0) = 8.0 m.

10) Select all valid rearrangements of height₁/shadow₁ = height₂/shadow₂.

Height₂ = (height₁·shadow₂)/shadow₁

Shadow₁ = (height₁·shadow₂)/height₂

Height₁ = (height₂·shadow₁)/shadow₂

Shadow₂ = (height₂·shadow₁)/height₁

Height₂ = (shadow₁·shadow₂)/height₁

Cross-multiplication gives A–D. Option E mixes units incorrectly.

Explanation

Cross-multiplication gives A–D. Option E mixes units incorrectly.

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11) Select all equations that correctly express the same-sun-angle relationship.

Height₁/shadow₁ = height₂/shadow₂

Height₁·shadow₂ = height₂·shadow₁

Height₂ = height₁·(shadow₂/shadow₁)

Shadow₂ = shadow₁·(height₁/height₂)

Height₂ = height₁·(shadow₁/shadow₂)

From height₁/shadow₁ = height₂/shadow₂, cross-multiply to get height₁·shadow₂ = height₂·shadow₁ and solve for any unknown.

Explanation

From height₁/shadow₁ = height₂/shadow₂, cross-multiply to get height₁·shadow₂ = height₂·shadow₁ and solve for any unknown.

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12) A 2.25 m marker casts a 1.10 m shadow. A statue’s shadow is 4.60 m. What is the statue’s height (to 2 d.p.)?

9.21 m

9.41 m

9.61 m

9.81 m

height₂ = 2.25·(4.60/1.10) = 2.25·4.1818… ≈ 9.41 m.

Explanation

height₂ = 2.25·(4.60/1.10) = 2.25·4.1818… ≈ 9.41 m.

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14) A 2.0 m stick casts a 1.6 m shadow. At the same time, a statue’s shadow is 2.4 m. What is the statue’s height?

3 m

2.6 m

3.4 m

3.8 m

height₂ = height₁·(shadow₂/shadow₁) = 2.0·(2.4/1.6) = 3.0 m.

Explanation

height₂ = height₁·(shadow₂/shadow₁) = 2.0·(2.4/1.6) = 3.0 m.

15) A 1.4 m traffic cone has a 0.9 m shadow. A sculpture’s shadow is 2.7 m. Find the sculpture height.

3.8 m

4.2 m

4.6 m

5.0 m

height₂ = 1.4·(2.7/0.9) = 4.2 m.

Explanation

height₂ = 1.4·(2.7/0.9) = 4.2 m.

16) If the sun angle stays the same but the ground tilts, the ratio method height/shadow still holds without adjustment.

True

False

The method assumes level ground and vertical objects. A slope changes the geometry.

Explanation

The method assumes level ground and vertical objects. A slope changes the geometry.

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18) A 1.8 m post has a 1.2 m shadow. A tree’s shadow is 4.0 m. Estimate the tree’s height.

5.4 m

5.8 m

6.0 m

6.6 m

height₂ = 1.8·(4.0/1.2) = 6.0 m.

Explanation

height₂ = 1.8·(4.0/1.2) = 6.0 m.

19) A 2.0 m pole has a 1.6 m shadow. A statue is 3.0 m tall at the same time. The statue’s shadow is ____ m.

shadow₂ = shadow₁·(height₂/height₁) = 1.6·(3.0/2.0) = 2.4 m.

Explanation

shadow₂ = shadow₁·(height₂/height₁) = 1.6·(3.0/2.0) = 2.4 m.

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20) If height is measured in centimeters and shadow in meters for the same object, the ratio must be converted to consistent units before comparing.

True

False

Make the ratio unitless by converting to common units.

Explanation

Make the ratio unitless by converting to common units.

Height From Shadow Quiz: Height from Shadow Using Ratio