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:

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  1. 50 m
  2. \(\frac{50\sqrt3}{3}\) m
  3. 50√3 m
  4. 75 m

Answer (Detailed Solution Below)

Option 3 : 50√3 m
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F1 Arun Madhuri 26.10.2021 D9

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

F1 Arun Madhuri 26.10.2021 D10

3 + 1 = 4 = 100 m

\(\Rightarrow 2√{3}= \frac{100}{4}\times 2√{3}\)

AC = 50√3 

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