Find the height of the tunnel.

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Solution

Q.
The Atal Tunnel, also known as the Rohtang Tunnel, is a highway tunnel built under the Rohtang Pass in the eastern Pir Panjal range of the Himalayas, located on the Leh–Manali highway in Himachal Pradesh. Spanning a length of 9.02 km, it is the world’s longest tunnel situated above 10,000 feet and is named after the former Prime Minister of India, Atal Bihari Vajpayee. The cross-section of a tunnel is shown in figure. The radius of circular part is $5 square root of 2 space end root m e t r e s space a n d angle A O B equals 90 degree.$

Based on the above information, answer the following questions:

(ii) Find the height of the tunnel.

1 $5 (\sqrt{2} + 3) m$

2 $4 (\sqrt{2} + 1) m$

3 $5 (\sqrt{2} + 1) m$

4 $5 (\sqrt{- 2} + 1) m$

Ans.
(3)

$T h e h e i g h t o f t h e t u n n e l w i l l b e (O M + O C) . I n ∆ O A M a n d ∆ O B M$

$∠ O A M = ∠ O B M (∵ O A = O B) ∠ O M A = ∠ O M B = 90 °$

$B y A A s i m i l a r i t y, ∆ O A M ~ ∆ O B M A M = M B = \frac{A B}{2} = 5 m$

$In right ∆ OMB : {OB}^{2} = {OM}^{2} + {MB}^{2}$

$(5 \sqrt{2})^{2} = {OM}^{2} + (5)^{2} 50 = {OM}^{2} + 25 OM = 5 m$

$OC = OA = 5 \sqrt{2} m ∴ Height of the tunnel = OM + OC = 5 (\sqrt{2} + 1) m$

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