Two-level observation: elevation and depression: A vertical tower subtends an angle of 30° at a ground-level point P. From a second point directly above P by h metres (same vertical line), the angle of depression to the foot of the tower is 60°. Find the horizontal distance from P to the tower in terms of h.
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Ah cot 60°
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Bh cot 30°
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Ch cot 60° 2
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Dh cot 30° 2
Answer
Correct Answer: h cot 60°
Explanation
Introduction / Context:One observation is an elevation from the ground; the other is a depression from a point vertically above it. The horizontal distance remains the same for both, allowing a direct relation to h via tangent of 60°.
Given Data / Assumptions:
- At P (ground), angle of elevation to top = 30°.
- At Q (directly above P by h), angle of depression to foot = 60°.
- Horizontal distance from P (and Q) to tower base = x (unknown).
Concept / Approach:From Q to the tower foot: tan 60° = vertical drop / horizontal = h / x → x = h / √3 = h cot 60°. The first observation would then set the tower height, but the question only asks x.
Step-by-Step Solution:
tan 60° = h / x → x = h / √3 = h cot 60°.Consistency check (optional): If tower height is T, tan 30° = T / x → T = x / √3 = h / 3, which is positive and consistent.Verification / Alternative check:Units: x and h are lengths; cot 60° is dimensionless; expression is dimensionally sound.
Why Other Options Are Wrong:h cot 30° equals h√3, not supported by tan 60° relation; the “squared” looking options are not meaningful here.
Common Pitfalls:Using elevation (30°) to compute x directly; mixing up which angle applies to which vertical difference; confusing depression with elevation.
Final Answer:h cot 60°