Angle doubles after moving forward: From a point 160 m from a vertical tower, the angle subtended by the tower is θ. After advancing 100 m straight toward the tower, the subtended angle becomes 2θ. Find the height of the tower.

Introduction / Context:
Here a single unknown height must satisfy a double-angle relation as the observer moves closer. Using tan identities links the two geometry situations.

Given Data / Assumptions:

• Initial horizontal distance = 160 m; after moving = 60 m.
• Initial angle = θ; new angle = 2θ.
• Height of tower = h.

Concept / Approach:
Set tan θ = h / 160 and tan 2θ = h / 60. Use tan double-angle: tan 2θ = 2 t / (1 - t^2) with t = tan θ. Solve for h.

Step-by-Step Solution:

Verification / Alternative check:
With h = 80, tan θ = 80/160 = 1/2. tan 2θ = 2*(1/2)/(1 - 1/4) = 1/(3/4) = 4/3. At 60 m, h/60 = 80/60 = 4/3 ✔️.

Why Other Options Are Wrong:
100, 160, 200 m fail the tan 2θ equality when checked; only 80 m satisfies both positions.

Common Pitfalls:
Forgetting that observer distance changes to 60 m; sign errors in the tan 2θ formula; algebraic mistakes when clearing denominators.

Final Answer:
80 m