Difficulty: Easy
Correct Answer: 30°
Explanation:
Introduction:
This conceptual trigonometry question links shadow length and object height to the Sun’s angle of elevation. When the ratio of height to shadow length is known, we can directly use the definition of tangent to find the angle. Recognising standard trigonometric ratios is very helpful here.
Given Data / Assumptions:
Concept / Approach:
In a right-angled triangle formed by the tower, its shadow, and the line of sight to the top, the tangent of the Sun’s altitude θ is:
tan θ = (opposite side) / (adjacent side) = height / shadow = h / L.We substitute L = √3 * h and simplify to identify which standard angle has this tangent.
Step-by-Step Solution:
Step 1: Use the formula: tan θ = h / L.Step 2: Substitute L = √3 * h: tan θ = h / (√3 * h).Step 3: Cancel h (assuming h ≠ 0): tan θ = 1 / √3.Step 4: Recall standard trigonometric values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.Step 5: Therefore, θ = 30°.
Verification / Alternative check:
We can quickly verify: for θ = 30°, tan 30° = 1/√3, so L = h / tan 30° = h / (1/√3) = √3 * h. This matches the given condition that the shadow length is √3 times the height of the tower.
Why Other Options Are Wrong:
At 45°, tan 45° = 1, giving L = h, not √3 * h. At 60°, tan 60° = √3, giving L = h / √3, which is shorter than the height. The option "none" is incorrect because a standard angle (30°) clearly satisfies the condition. 75° has a tangent significantly greater than 1, also contradicting the given ratio.
Common Pitfalls:
A common error is to invert the ratio and treat tan θ as L/h instead of h/L, which leads to the wrong angle. Another mistake is misremembering standard tangent values. To avoid confusion, always set up tan θ = opposite / adjacent with height as opposite and shadow as adjacent, then match the ratio to known values.
Final Answer:
The angle of elevation of the Sun is 30°.
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