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

Options

A. 800 m
B. 700 m
C. 900 m
D. 1000 m

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have two trains. The first train travels at a speed of 36 km/hr and passes another train (which is half its length) in 12 seconds. The second train travels at 54 km/hr in the opposite direction. The first train also passes a railway platform in 1.5 minutes (or 90 seconds). We need to find the length of the platform.

### Step 2: Convert Speeds to m/s
To work with the speeds more easily, we will convert them from km/hr to m/s.
- Speed of the first train:
\[ 36 \text{ km/hr} = \frac{36 \times 1000}{3600} = 10 \text{ m/s} \]
- Speed of the second train:
\[ 54 \text{ km/hr} = \frac{54 \times 1000}{3600} = 15 \text{ m/s} \]

### 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: Calculate the Length of the First Train
Let the length of the first train be \( x \) meters. The length of the second train, which is half of the first train, will be \( \frac{x}{2} \) meters.

The total distance covered when both trains pass each other is the sum of their lengths:
\[ \text{Distance} = x + \frac{x}{2} = \frac{3x}{2} \]

Using the formula for distance (Distance = Speed × Time):
\[ \text{Distance} = 25 \text{ m/s} \times 12 \text{ s} = 300 \text{ m} \]
Setting the two distance equations equal gives us:
\[ \frac{3x}{2} = 300 \]
To solve for \( x \):
\[ 3x = 600 \implies x = 200 \text{ m} \]
So, the length of the first train is 200 meters, and the length of the second train is:
\[ \frac{200}{2} = 100 \text{ m} \]

### Step 5: Calculate the Length of the Platform
Now, we know the length of the first train is 200 meters. The first train passes a platform in 90 seconds. Let the length of the platform be \( y \) meters.

Using the distance formula again:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
The distance covered while passing the platform is the length of the train plus the length of the platform:
\[ 200 + y = 10 \text{ m/s} \times 90 \text{ s} = 900 \text{ m} \]
Now, we can solve for \( y \):
\[ 200 + y = 900 \implies y = 900 - 200 = 700 \text{ m} \]

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

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