Approaching a tower — elevation changes from 30° to 60° in 20 m On level ground, the angle of elevation of a tower is 30°. After walking 20 m toward the tower, the angle becomes 60°. Find the tower’s height.

Introduction / Context:
Two observations from points along the same line to a tower give two tan equations in the same unknowns (height and the initial horizontal distance). Subtracting distances resolves both quickly.

Given Data / Assumptions:

• Initial angle θ₁ = 30°, then θ₂ = 60° after moving 20 m closer.
• Let initial distance be x, tower height h.
• Level ground; vertical tower.

Concept / Approach:
tan 30° = h/x and tan 60° = h/(x − 20). Solve for x, then h. Use tan 30° = 1/√3 and tan 60° = √3.

Step-by-Step Solution:

Verification / Alternative check:
At 30 m, tan 30° = h/x = (10√3)/30 = 1/√3; at 10 m (after moving 20), tan 60° = (10√3)/10 = √3. Checks out.

Why Other Options Are Wrong:
20√3 is double the correct height; 10(√3 − 1) and “None” do not satisfy both tan equations.

Common Pitfalls:
Using 20 as the new distance (x − 20 = 20) without solving, or mixing sine/cosine for tangent relations.

Final Answer:
10 √3 m