Question

Two lamp posts are of equal heights. A boy measured the elevation of the top of each lamp post from the midpoint of the line segment joining the feet of the lamp post, as 30°. After walking 15 m towards one of them, he measured the elevation of the top of the nearest lamp post at the point where he stands as 60°.

Based on the above information, answer the following questions.

1. Taking h metres as the height of each lamp post and x metres as the distance between the feet of the two lamp posts, find a relation between x and h.
2. After moving 15 m towards one lamp post, what is the relation between x and h?
3. Find the height of each lamp post.
4. Find the distance between the two lamp posts.

Answer 1

Let the lamp posts be AB and CD of equal height h metres, with feet B and D respectively. Let E be the midpoint of BD. Then \(BE = ED = \frac{x}{2}\) metres and \(\angle AEB = \angle CED = 30^\circ\).

i. Relation between x and h

In right \(\triangle CDE\): \[\tan 30^\circ = \frac{CD}{ED} = \frac{h}{x/2}\]

\[\Rightarrow \frac{1}{\sqrt{3}} = \frac{2h}{x}\]

\[\Rightarrow x = 2\sqrt{3}h\]

ii. Relation after moving 15 m towards one lamp post

Let the boy move 15 m towards lamp post CD to a point F. Then EF = 15 m and \(FD = \left(\frac{x}{2} - 15\right)\) m.

In right \(\triangle CDF\): \[\tan 60^\circ = \frac{CD}{FD} = \frac{h}{\frac{x}{2} - 15}\]

\[\Rightarrow \sqrt{3} = \frac{2h}{x - 30}\]

\[\Rightarrow 2h = \sqrt{3}(x - 30)\]

iii. Height of each lamp post

Substitute \(x = 2\sqrt{3}h\) from (i) into the relation from (ii):

\[2h = \sqrt{3}(2\sqrt{3}h - 30)\]

\[\Rightarrow 2h = 6h - 30\sqrt{3}\]

\[\Rightarrow 4h = 30\sqrt{3}\]

\[\Rightarrow h = \frac{15\sqrt{3}}{2} \text{ m}\]

iv. Distance between the two lamp posts

From (i), \(x = 2\sqrt{3}h\).

Substituting \(h = \frac{15\sqrt{3}}{2}\): 

\[x = 2\sqrt{3} \times \frac{15\sqrt{3}}{2}\]

\[= 45 \text{ m}\]