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.

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:

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°