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

Options

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

Answer

To solve the problem, we need to find the length \( L \) of the tunnel given the lengths of the trains and their speeds. Let's break down the solution step by step.

### Step 1: Convert Speeds to m/s
The speeds of the trains are given in km/h. We need to convert these speeds to m/s for easier calculations.

- Speed of Train A:
\[ 108 \text{ km/h} = \frac{108 \times 1000}{3600} = 30 \text{ m/s} \]

- Speed of Train B:
\[ 72 \text{ km/h} = \frac{72 \times 1000}{3600} = 20 \text{ m/s} \]

### Step 2: Calculate the Time Taken by Each Train to Cross the Tunnel
The time taken by each train to cross the tunnel can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

- For Train A:
The distance Train A needs to cover is the length of the tunnel \( L \) plus its own length \( l \):
\[ \text{Time}_A = \frac{L + l}{30} \]

- For Train B:
The distance Train B needs to cover is the length of the tunnel \( L \) plus its own length \( 4l \):
\[ \text{Time}_B = \frac{L + 4l}{20} \]

### Step 3: Set Up the Equation Based on the Given Information
According to the problem, Train A takes 35 seconds less than Train B to cross the tunnel:
\[ \text{Time}_B - \text{Time}_A = 35 \]
Substituting the expressions for time:
\[ \frac{L + 4l}{20} - \frac{L + l}{30} = 35 \]

### Step 4: Find a Common Denominator and Simplify
The common denominator for 20 and 30 is 60. We can rewrite the equation:
\[ \frac{3(L + 4l)}{60} - \frac{2(L + l)}{60} = 35 \]
Combining the fractions:
\[ \frac{3(L + 4l) - 2(L + l)}{60} = 35 \]
Expanding the numerator:
\[ \frac{3L + 12l - 2L - 2l}{60} = 35 \]
This simplifies to:
\[ \frac{L + 10l}{60} = 35 \]

### Step 5: Solve for \( L \)
Multiplying both sides by 60:
\[ L + 10l = 2100 \]
Rearranging gives:
\[ L = 2100 - 10l \]

### Step 6: Use the Given Relation \( L = 60l \)
We know from the problem that \( L = 60l \). Substituting this into the equation:
\[ 60l = 2100 - 10l \]
Combining like terms:
\[ 70l = 2100 \]
Dividing both sides by 70:
\[ l = 30 \]

### Step 7: Find \( L \)
Now substituting \( l \) back into the equation for \( L \):
\[ L = 60l = 60 \times 30 = 1800 \]

### Final Answer
The length of the tunnel \( L \) is:
\[ \boxed{1800 \text{ meters}} \]