Question

A line segment AB of length 10 meters is passing through the foot of the perpendicular of a pillar, which is standing at right angle to the ground. The tip of the pillar subtends angles \( \tan^{-1}(3) \) and \( \tan^{-1}(2) \) at A and B respectively. Which of the following choices represents the height of the pillar?

• A. 10 meters
• B. 8 meters
• C. 12 meters
• D. 15 meters

Hint:

When solving problems involving angles of elevation and horizontal distances, use the tangent function to relate the height of the object and the distances from the observation points.

Answer 1

The Correct Option is C

Let the height of the pillar be \( h \). Since the angles \( \tan^{-1}(3) \) and \( \tan^{-1}(2) \) represent the angles subtended by the pillar at points A and B, we can use the tangent of these angles to form two equations. 1. At point A, the angle subtended is \( \tan^{-1}(3) \), so: \[ \tan \left( \tan^{-1}(3) \right) = 3 \] This means: \[ \frac{h}{d_1} = 3 \Rightarrow d_1 = \frac{h}{3} \text{(1)} \] where \( d_1 \) is the horizontal distance from point A to the foot of the pillar. 2. At point B, the angle subtended is \( \tan^{-1}(2) \), so: \[ \tan \left( \tan^{-1}(2) \right) = 2 \] This means: \[ \frac{h}{d_2} = 2 \Rightarrow d_2 = \frac{h}{2} \text{(2)} \] where \( d_2 \) is the horizontal distance from point B to the foot of the pillar. Now, we are given that the length of the line segment AB is 10 meters. So: \[ d_1 + d_2 = 10 \] Substituting the expressions for \( d_1 \) and \( d_2 \) from equations (1) and (2): \[ \frac{h}{3} + \frac{h}{2} = 10 \] To solve for \( h \), find a common denominator: \[ \frac{2h}{6} + \frac{3h}{6} = 10 \] \[ \frac{5h}{6} = 10 \] \[ h = \frac{10 \times 6}{5} = 12 \] Thus, the height of the pillar is \( \boxed{12} \) meters.