Height From Shadow Quiz: Height from Shadow Using Ratio

Option A is the direct proportion. Option B is its cross-multiplied form. Option C solves for height₂ correctly by dividing both sides by shadow₁. Option D states shadow₂ = shadow₁×(height₁/height₂), which inverts the height ratio — the correct form is shadow₂ = shadow₁×(height₂/height₁).

The answer is True. The proportion height₁/shadow₁ = height₂/shadow₂ requires all four measurements to use the same unit so the ratios are dimensionally consistent. If height is in centimeters and shadow is in meters, the ratio height/shadow equals centimeters per meter, which is 100 times larger than the correct dimensionless ratio. Comparing this inflated ratio to a correctly computed ratio from another object produces an incorrect result. Converting to a common unit before calculating eliminates this error.

shadow₂ = shadow₁×(height₂/height₁) = 1.6×(3.0/2.0) = 1.6×1.5 = 2.4 m. Option A gives 2.0, requiring height₂/height₁ = 1.25, meaning height₂ = 2.5 m, not 3.0 m. Option C gives 2.8, requiring the ratio to equal 1.75. Option D gives 3.2, requiring 2.0. Only 2.4 m satisfies the proportion with height₂/height₁ = 1.5.

height₂ = 1.8×(4.0/1.2) = 1.8×3.333 = 6.0 m. Option A gives 5.4, requiring shadow₂/shadow₁ = 3.0, meaning shadow₂ = 3.6 m, not 4.0 m. Option B gives 5.8, requiring the ratio to equal 3.222. Option D gives 6.6, requiring 3.667. Only 6.0 m satisfies the proportion with shadow₂/shadow₁ = 3.333.

height₂ = 3×(7/2) = 3×3.5 = 10.5 m. Option A gives 9.5, requiring shadow₂/shadow₁ = 3.167, meaning shadow₂ = 6.33 m, not 7 m. Option C gives 11.5, requiring the ratio to equal 3.833. Option D gives 12.0, requiring 4.0. Only 10.5 m satisfies the proportion with shadow₂/shadow₁ = 3.5.

The answer is False. The shadow ratio method assumes level ground so that the shadow falls on a flat horizontal surface. When the ground tilts, the shadow length changes depending on the slope direction and angle, altering the geometry of the right triangle. A slope toward the sun shortens the shadow and a slope away lengthens it, meaning the measured shadow no longer equals the horizontal distance used in tanθ = height/shadow. An adjustment for slope is required.

height₂ = 1.4×(2.7/0.9) = 1.4×3 = 4.2 m. Option A gives 3.8, requiring shadow₂/shadow₁ = 2.714, meaning shadow₂ = 2.44 m, not 2.7 m. Option C gives 4.6, requiring the ratio to equal 3.286. Option D gives 5.0, requiring 3.571. Only 4.2 m satisfies the proportion with shadow₂/shadow₁ = 3.

height₂ = 2.0×(2.4/1.6) = 2.0×1.5 = 3.0 m. Option B gives 2.6, requiring shadow₂/shadow₁ = 1.3, meaning shadow₂ = 2.08 m, not 2.4 m. Option C gives 3.4, requiring the ratio to equal 1.7. Option D gives 3.8, requiring 1.9. Only 3.0 m satisfies the proportion with shadow₂/shadow₁ = 1.5.

height₂ = 0.9×(1.1/0.6) = 0.9×1.8333 = 1.65 m. Option A gives 1.35, requiring shadow₂/shadow₁ = 1.5, meaning shadow₂ = 0.9 m, not 1.1 m. Option B gives 1.50, requiring the ratio to equal 1.667. Option D gives 1.80, requiring 2.0. Only 1.65 m satisfies the proportion with shadow₂/shadow₁ = 1.8333.

height₂ = 2.25×(4.60/1.10) = 2.25×4.1818 = 9.409 ≈ 9.41 m. Option A gives 9.21, requiring 4.60/1.10 = 4.093, meaning shadow₁ = 1.124 m, not 1.10 m. Option C gives 9.61, requiring the ratio to equal 4.271. Option D gives 9.81, requiring 4.36. Only 9.41 m satisfies the proportion with the given measurements.

height₂ = height₁×(shadow₂/shadow₁) = 2.75×(3.00/1.25) = 2.75×2.4 = 6.60 m. Option A gives 5.80, requiring shadow₂/shadow₁ = 2.109, meaning shadow₂ = 2.636 m, not 3.00 m. Option C gives 7.20, requiring the ratio to equal 2.618. Option D gives 4.95, requiring 1.8. Only 6.60 m satisfies the proportion with the given values.

Cross-multiplying height₁/shadow₁ = height₂/shadow₂ gives height₁×shadow₂ = height₂×shadow₁. Option A solves for height₂, option B solves for shadow₁, option C solves for height₁, and option D solves for shadow₂ — all are valid rearrangements. 

height₂ = 3.2×(5.0/2.0) = 3.2×2.5 = 8.0 m. Option A gives 7.6, requiring shadow₂/shadow₁ = 2.375, meaning shadow₂ = 4.75 m, not 5.0 m. Option C gives 8.4, requiring the ratio to equal 2.625. Option D gives 9.0, requiring 2.8125. Only 8.0 m satisfies the proportion with shadow₂/shadow₁ = 2.5.

The answer is True. Starting from height₁/shadow₁ = height₂/shadow₂, cross-multiplying gives height₁×shadow₂ = height₂×shadow₁. Dividing both sides by height₁ gives shadow₂ = shadow₁×(height₂/height₁). This is a valid algebraic rearrangement of the original proportion and allows shadow₂ to be found whenever height₁, height₂, and shadow₁ are known.

The proportion height₁/shadow₁ = height₂/shadow₂ has four variables. Any three of the four determine the fourth uniquely. Option A provides three values and solves for height₂ = height₁×(shadow₂/shadow₁). Option B provides three values and solves for shadow₂ = shadow₁×(height₂/height₁). Option C gives only two values, leaving two unknowns. Option D gives only shadow measurements with no height information. 

Option A inverts the ratio, producing the reciprocal of the correct answer. Option B changes the numerical value of one shadow relative to the other, breaking the proportion. Option C violates the equal sun angle assumption since tanθ changes with time. Option D uses a longer measurement than the true vertical height, inflating the result.

height₂ = 1.5×(3.6/1.2) = 1.5×3 = 4.50 m. Option A gives 3.80, requiring shadow₂/shadow₁ = 2.533, meaning shadow₂ = 3.04 m, not 3.6 m. Option B gives 4.10, requiring the ratio to equal 2.733. Option D gives 5.20, requiring 3.467. Only 4.50 m satisfies the proportion with shadow₂/shadow₁ = 3.

Option A causes failure because the method assumes perfectly vertical objects. Option B causes failure because sloped ground changes the shadow geometry. Option C causes failure because a changed sun angle means the equal-ratio assumption no longer holds between the two objects. Option D is not a failure condition — measuring both objects simultaneously on level ground is exactly the correct setup for the shadow ratio method to work accurately.

The answer is True. At any given moment, the sun's angle of elevation θ is the same for all objects on level ground. Since tanθ = height/shadow, every vertical object satisfies the same ratio. This means height₁/shadow₁ = height₂/shadow₂ = tanθ, making the ratios equal. This equality is the geometric foundation of the shadow ratio method for estimating unknown heights.

The answer is True. The proportion is height₁/shadow₁ = height₂/shadow₂. If both shadows double, the new proportion becomes height₁/(2×shadow₁) = height₂/(2×shadow₂). The factor of 2 cancels on both sides, leaving the original proportion unchanged. The computed heights are therefore identical to what they would have been before the shadows doubled, confirming the ratio method is unaffected.