Question

A tower \( A \) leans towards west making an angle \( \theta \) with the vertical. The angular elevation of \( B \), the topmost point of the tower is \( \beta \) as observed from a point \( C \) at a distance \( d' \) from \( B \). If the angular elevation of \( B \) from point \( D \) due east of \( C \) is the same and \( 2d \) from \( C \), then \( \theta \) can be given as

• A. \( \tan \theta = \frac{2}{3} \)
• B. \( \tan \theta = \frac{3}{2} \)
• C. \( \tan \theta = \frac{1}{2} \)
• D. \( \tan \theta = \frac{1}{3} \)

Hint:

Use trigonometric identities and geometry to solve problems involving angles of elevation and depression.

Answer 1

The Correct Option is C

Step 1: Analyzing the geometry.
The relationship between the height of the tower, the angles of elevation, and the distance from the point of observation can be used to derive the tangent of \( \theta \). Using trigonometry, \( \tan \theta = \frac{1}{2} \).

Step 2: Conclusion.
The correct value of \( \tan \theta \) is \( \frac{1}{2} \), corresponding to option (3).