Two successive elevations after walking a: From a point A, the angle of elevation of the top of a vertical tower is α. After walking a metres straight toward the tower, the angle becomes β (β > α). Express the height of the tower in terms of a, α, β.
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Aa Sin α Sin β/ Sin(β - α)
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Ba Sin α Sin β/Sin(α - β)
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Ca Sin( β - α ) Sin α Sin β
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Da Sin(α - β)/Sin αSin β
Answer
Correct Answer: a Sin α Sin β/ Sin(β - α)
Explanation
Introduction / Context:This is the standard two-elevation formula for a vertical tower when an observer moves a known distance a toward the tower, changing the angle from α to β. A sine-rule derivation yields a compact expression for the height.
Given Data / Assumptions:
- Initial elevation = α; after walking a toward tower, elevation = β.
- Height of tower = H.
- Both observations on the same straight line toward the tower.
Concept / Approach:Let initial horizontal distance be x. Then tan α = H / x and tan β = H / (x − a). Eliminating x gives H = a * tan α * tan β / (tan β − tan α). Using the sine addition identity, this equals a * sin α * sin β / sin(β − α).
Step-by-Step Solution:
tan α = H / x → x = H / tan α.tan β = H / (x − a) → x − a = H / tan β.Subtract: a = H(1/ tan α − 1/ tan β) = H( (tan β − tan α)/(tan α tan β) ).Thus H = a * tan α * tan β / (tan β − tan α) = a * sin α * sin β / sin(β − α).Verification / Alternative check:Plug small numeric values (e.g., α = 30°, β = 60°, a = known) to confirm both tan-form and sine-form give identical H.
Why Other Options Are Wrong:They have wrong sign in denominator, extra factors, or inverted ratios; only option (a) matches the proven identity.
Common Pitfalls:Using degrees with calculator in radian mode; mixing tan-based and sin-based expressions mid-derivation; forgetting that β > α so denominator is positive.
Final Answer:a Sin α Sin β/ Sin(β - α)