A train travelling at 36 km/hr. completely crosses another train having half its length and travelling in the opposite direction at 54 km/hr. in 12 seconds. If it also passes a railway plat form in `1(1)/(2)` minutes, the length of the platform is:

Options

A. 740 metres
B. 700 metres
C. 620 metres
D. 560 metres

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We have two trains moving in opposite directions. The first train travels at 36 km/hr and the second train travels at 54 km/hr. The first train completely crosses the second train in 12 seconds. We need to find the length of a platform that the first train crosses in 1.5 minutes.

### Step 2: Convert Speeds to Meter per Second
To work with the speeds in a more manageable unit (meters per second), we convert the speeds from kilometers per hour to meters per second using the conversion factor \( \frac{5}{18} \).

- Speed of the first train:
\[ 36 \text{ km/hr} = 36 \times \frac{5}{18} = 10 \text{ m/s} \]

- Speed of the second train:
\[ 54 \text{ km/hr} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 3: Calculate the Relative Speed
Since the trains are moving in opposite directions, we add their speeds to find the relative speed:
\[ \text{Relative Speed} = 10 \text{ m/s} + 15 \text{ m/s} = 25 \text{ m/s} \]

### Step 4: Set Up the Equation for Crossing the Second Train
Let the length of the first train be \( 2x \) and the length of the second train be \( x \) (since it is half the length of the first train). The total distance covered when the first train crosses the second train is:
\[ \text{Distance} = 2x + x = 3x \]
Using the formula for time, which is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
We can set up the equation:
\[ 12 \text{ seconds} = \frac{3x}{25} \]

### Step 5: Solve for \( x \)
Rearranging the equation gives:
\[ 3x = 12 \times 25 \]
\[ 3x = 300 \]
\[ x = 100 \text{ meters} \]
Thus, the length of the first train is:
\[ 2x = 2 \times 100 = 200 \text{ meters} \]

### Step 6: Calculate the Time Taken to Cross the Platform
The first train crosses a platform in 1.5 minutes. Converting this time to seconds:
\[ 1.5 \text{ minutes} = 1.5 \times 60 = 90 \text{ seconds} \]

### Step 7: Calculate the Distance Covered While Crossing the Platform
Using the speed of the first train (which is 10 m/s), we can find the total distance covered while crossing the platform:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 10 \text{ m/s} \times 90 \text{ s} = 900 \text{ meters} \]

### Step 8: Find the Length of the Platform
The distance covered while crossing the platform is the sum of the lengths of the first train and the platform:
\[ \text{Distance} = \text{Length of the first train} + \text{Length of the platform} \]
Thus,
\[ 900 = 200 + \text{Length of the platform} \]
Solving for the length of the platform gives:
\[ \text{Length of the platform} = 900 - 200 = 700 \text{ meters} \]

### Final Answer
The length of the platform is **700 meters**.

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