Question

Two vertical lamp-posts of equal height stand on either side of a road 50 m wide. At a point \(P\) on the road between them, the elevation of the tops of the lamp-posts are \(60^\circ\) and \(30^\circ\). Find the distance of \(P\) from the lamp-post which makes angle of \(60^\circ\).

• A. 25 m
• B. 12.5 m
• C. 16.5 m
• D. 20.5 m

Hint:

Set up two tangent equations for the two angles of elevation and equate heights to solve for distance.

Answer 1

The Correct Option is B

Let the height of each lamp-post be \(h\). Let \(x\) be the distance from \(P\) to the \(60^\circ\) lamp-post. From \(60^\circ\) post: \[ \tan 60^\circ = \frac{h}{x} \Rightarrow h = x\sqrt{3} \] From \(30^\circ\) post at distance \(50 - x\): \[ \tan 30^\circ = \frac{h}{50 - x} \Rightarrow h = \frac{50 - x}{\sqrt{3}} \] Equating: \[ x\sqrt{3} = \frac{50 - x}{\sqrt{3}} \Rightarrow 3x = 50 - x \Rightarrow x = 12.5 \] Thus, \(P\) is \( {12.5\ \text{m}} \) from the \(60^\circ\) lamp-post.