A 270-m long train takes 24 seconds to cross a pole. If it takes 1 minute to cross a tunnel, then the length (in m ) of the tunnel is :

Options

A. 507
B. 405
C. 408
D. 510

Answer

To find the length of the tunnel that the train crosses, we can follow these steps:

### Step 1: Calculate the speed of the train.
The speed of the train can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Here, the distance is the length of the train, which is 270 meters, and the time taken to cross a pole is 24 seconds.

\[ \text{Speed} = \frac{270 \text{ m}}{24 \text{ s}} = \frac{270}{24} \text{ m/s} \]

### Step 2: Simplify the speed calculation.
To simplify \( \frac{270}{24} \):
- Divide both the numerator and the denominator by 3:
\[ \frac{270 \div 3}{24 \div 3} = \frac{90}{8} \]
- Now, divide both by 2:
\[ \frac{90 \div 2}{8 \div 2} = \frac{45}{4} \text{ m/s} \]

### Step 3: Set up the equation for crossing the tunnel.
When the train crosses the tunnel, it travels a distance equal to its own length plus the length of the tunnel (let's denote the length of the tunnel as \( X \)):
\[ \text{Distance} = 270 + X \]

The time taken to cross the tunnel is given as 1 minute, which is equal to 60 seconds.

### Step 4: Use the speed to find the length of the tunnel.
Using the speed we calculated, we can set up the equation:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Substituting the known values:
\[ \frac{45}{4} = \frac{270 + X}{60} \]

### Step 5: Solve for \( X \).
Cross-multiplying gives:
\[ 45 \times 60 = 4 \times (270 + X) \]
Calculating \( 45 \times 60 \):
\[ 2700 = 4 \times (270 + X) \]
Now, divide both sides by 4:
\[ 675 = 270 + X \]
Subtract 270 from both sides:
\[ X = 675 - 270 = 405 \]

### Conclusion:
The length of the tunnel is \( 405 \) meters.

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