Question

Two trains 'A' and 'B' of length \('l'\) and \('4l'\) are travelling into a tunnel of length 'L' in parallel tracks from opposite directions with velocities 108 km/h and 72 km/h, respectively. If train 'A' takes 35s less time than train 'B' to cross the tunnel then. length 'L' of tunnel is:
(Given L = 60 \(l\))

• A. 1800 m
• B. 900 m
• C. 1200 m
• D. 2700 m

Hint:

When two trains are moving in opposite directions, the relative speed is the sum of their individual speeds. Use this concept to determine the time difference when passing through the same tunnel.

Answer 1

The Correct Option is A

To solve this problem, we need to determine the length of the tunnel 'L'. Given that train 'A' and train 'B' have lengths \(l\) and 4l\) respectively, they are traveling into the tunnel from opposite directions with velocities of 108 km/h and 72 km/h. We know the formula to find the time taken to cross a certain distance is:

t = \frac{\text{Distance}}{\text{Speed}}

First, convert the velocities from km/h to m/s:

• Velocity of Train 'A': 108 \text{ km/h} = \frac{108 \times 1000}{3600} = 30 \text{ m/s}
• Velocity of Train 'B': 72 \text{ km/h} = \frac{72 \times 1000}{3600} = 20 \text{ m/s}

The time taken by Train 'A' to cross the tunnel is:

t_A = \frac{L + l}{30}

The time taken by Train 'B' to cross the tunnel is:

t_B = \frac{L + 4l}{20}

According to the problem, Train 'A' takes 35 seconds less than Train 'B' to cross the tunnel:

t_B - t_A = 35

Substitute the expressions for t_A and t_B:

\frac{L + 4l}{20} - \frac{L + l}{30} = 35

To solve this equation, find a common denominator and simplify:

\frac{3(L + 4l) - 2(L + l)}{60} = 35

Simplify the equation further:

\frac{3L + 12l - 2L - 2l}{60} = 35

\frac{L + 10l}{60} = 35

Now, multiply both sides by 60:

L + 10l = 2100

We know from given information that L = 60l. Substitute L = 60l in the equation:

60l + 10l = 2100

70l = 2100

Solving for l:

l = \frac{2100}{70} = 30

So, the length of the tunnel L = 60 \times 30 = 1800 \text{ m}.

Thus, the correct answer is 1800 m.