Two observations from one line: From a point G on level ground, the angle of elevation of a vertical tower is 30°. After walking 20 m straight towards the tower, the angle of elevation becomes 60°. Find the height of the tower.

Difficulty: Medium

10 m √3
20 √3 m
20 m √3
10 √3 m

Correct Answer: 10 √3 m

Explanation:

Introduction / Context:
Height from two angles of elevation along the same straight line is a classic tangent setup using two right triangles with a common height and different adjacents.

Given Data / Assumptions:

Initial angle = 30° at distance x from base.
After moving 20 m closer, angle = 60° at distance x - 20.
Tower is vertical; ground is horizontal.

Concept / Approach:
Use tan(theta) = height / horizontal distance for each observation and equate the tower height expressions to solve for x, then compute height.

Step-by-Step Solution:

Let height = H, initial horizontal = x.tan 30° = H / x → H = x / √3.tan 60° = H / (x - 20) → H = √3 (x - 20).Equate: x / √3 = √3 (x - 20) → x = 3x - 60 → 2x = 60 → x = 30.Thus H = x / √3 = 30 / √3 = 10√3 m.

Verification / Alternative check:
With H = 10√3 and x = 30, second distance is 10; tan 60° = (10√3)/10 = √3 ✔️.

Why Other Options Are Wrong:
Values like 20√3 lead to inconsistent distances; only 10√3 satisfies both equations simultaneously.

Common Pitfalls:
Using 30° with (x - 20) instead of x; mixing up tan and cot; arithmetic slips when isolating x.

Two observations from one line: From a point G on level ground, the angle of elevation of a vertical tower is 30°. After walking 20 m straight towards the tower, the angle of elevation becomes 60°. Find the height of the tower.

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