Angle of elevation changes at the same observation point: From a point 120 m from the base of an unfinished vertical tower, the angle of elevation of the top is 45°. If, after raising the tower, the angle of elevation from the same point must become 60°, by how much height (in metres) should the tower be increased?

Introduction / Context:This problem uses right-triangle trigonometry for angles of elevation from a fixed horizontal distance. The initial angle is 45°; after increasing the tower’s height, the angle becomes 60°.

Given Data / Assumptions:

• Distance from observation point to tower base = 120 m (horizontal).
• Current angle of elevation = 45°.
• Target angle of elevation = 60°.
• Tower is vertical; ground is horizontal; observer’s eye level taken at ground level.

Concept / Approach:For a right triangle, tan(theta) = opposite / adjacent. Here, opposite = tower height as seen, adjacent = 120 m. Compute present height, compute required height, take the difference.

Step-by-Step Solution:

Verification / Alternative check:If raised by 120 (√3 - 1), the new total becomes 120√3. With adjacent 120, tan = (120√3)/120 = √3 → 60°, consistent.

Why Other Options Are Wrong:

• 120(√3 + 1): adds extra 120 beyond what is needed.
• 10(√3 + 1): wrong scale; ignores given distance.
• None of these: incorrect since a correct numeric form exists.

Common Pitfalls:Using sine or cosine instead of tangent; forgetting that only the increase is asked (not the final height).

Final Answer:120 ( √3 - 1 ) m