Depression to tower top and bottom from a cliff: From the top of a cliff of height h metres, the angles of depression of the top and the bottom of a nearby vertical tower are 30° and 60° respectively. Find the height of the tower in terms of h.

Introduction / Context:Two angles of depression from a higher point to two points on or above the ground allow expressing horizontal distance and relative verticals, yielding the unknown tower height as a function of h.

Given Data / Assumptions:

• Cliff height = h.
• Angles of depression: 30° to tower top, 60° to tower bottom.
• Tower is vertical; ground is level.

Concept / Approach:If horizontal distance from cliff foot to tower base is x, then tan 60° = h / x → x = h/√3. For the top, vertical drop is h - T (T = tower height). tan 30° = (h - T)/x → solve for T.

Step-by-Step Solution:

Verification / Alternative check:Insert T back: Top drop = h - 2h/3 = h/3 → tan 30° = (h/3)/(h/√3) = 1/√3 ✔️.

Why Other Options Are Wrong:h√3, 2h√3, h/3 mismatch the tangent relations; only 2h/3 satisfies both angles simultaneously.

Common Pitfalls:Confusing elevation with depression; misplacing T in h − T; forgetting that both rays share the same horizontal distance x.

Final Answer:2h/3