Question

Two pillars are of equal height and on either sides of a road, which is 100 m wide. The angles of elevation of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar.

Answer 1

Let AB and ED be two pillars each of height h metres. Let C be a point on the road such that BC = x metres.

Then CD = (100 – x) metres. Given, ∠ACB = 60°and ∠ECD = 30° 

In ΔABC, we have tan 60° = \(\frac{AB}{BC}\) ⇒ √3 = \(\frac{h}{x}\) ⇒ h = √3 x   .......(i)

In ΔECD, we have tan 30° = \(\frac{ED}{CD}\) = \(\frac{1}{\sqrt3}\) = \(\frac{h}{100-x}\) ⇒ h√3 = 100 -x  ...(ii)

From eq. (i) and (ii) on eliminating h, we have 

3\(x\) = 100 – \(x\) ⇒ 4\(x\) = 100 ⇒ \(x\) = 25 

Substituting \(x\) = 25 in (i), we have h = 25√3 = 25 × 1.732 = 43.3 m. 

Thus the required point is at a distance of 25 m from the first pillar and 75 m from the second pillar. The height of the pillars in 43.3 m.