Two walls of equal height are on either side of a road, which is 100 m wide. At a point on the road two stairs are along the two walls in such a way that they indicate the two angles of elevation from that point at 60° and 30°. The length of the longer stair is:

Given:

Distance between two walls = 100 m 

Concept:

The longer stair will be AC as the corresponding side making the smaller angle is larger than that of making bigger angle. 

Calculation:

Let the height of each of the walls = h 

BC + CE = BE = 100 

⇒ CE = 100 - BC .......(1) 

In triangle ABC, AB/BC = tan 30° 

⇒ h/BC = 1/√3 

⇒ BC = √3h.......(2)

In triangle CDE, DE/CE = tan 60° 

⇒ h/(100 - BC) = √3 

h/(100 - √3h) = √3 

h = 100√3 - 3h

4h = 100√3 

h = 25√3 m  ......  (3) 

By (2) and (3)

BC = √3h = √3 × 25√3

⇒ BC = 75 m 

Again in triangle ABC, BC/AC = cos 30° 

⇒ 75/AC = √3/2 

⇒ 150 = √3 × AC 

⇒ AC = 150/√3

⇒ AC = 150/√3 ×√3/√3

∴ AC = 50√3 m 

Alternate MethodBy Ratio Method

3 + 1 = 4 = 100 m

AC = 50√3