Speed of 2 trains are 45 km/hr and 42 km/hr both are running in opposite direction length of 1st train is double than 2nd. 1st train crosses a platform in 45 sec. and Both train crosses each other in 12 sec. then find the length of platform.

Options

A. 200 m
B. 400 m
C. 100 m
D. 300 m

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Define Variables
Let the length of the second train be \( x \) meters. Therefore, the length of the first train will be \( 2x \) meters.

### Step 2: Convert Speeds from km/hr to m/s
The speeds of the trains are given in km/hr. We need to convert them to m/s using the conversion factor \( \frac{5}{18} \).
- Speed of the first train:
\[ 45 \text{ km/hr} = 45 \times \frac{5}{18} = 12.5 \text{ m/s} \]
- Speed of the second train:
\[ 42 \text{ km/hr} = 42 \times \frac{5}{18} = 11.67 \text{ m/s} \]

### Step 3: Calculate the Combined Speed of Both Trains
Since the trains are moving in opposite directions, their speeds add up:
\[ \text{Combined Speed} = 12.5 + 11.67 = 24.17 \text{ m/s} \]

### Step 4: Use the Time Taken to Cross Each Other
The time taken for both trains to cross each other is given as 12 seconds. The distance covered when they cross each other is equal to the sum of their lengths:
\[ \text{Distance} = \text{Combined Speed} \times \text{Time} = 24.17 \times 12 = 290 \text{ meters} \]
This distance is equal to the length of both trains:
\[ 2x + x = 290 \implies 3x = 290 \implies x = \frac{290}{3} \approx 96.67 \text{ meters} \]

### Step 5: Calculate the Length of the First Train
Now, we can find the length of the first train:
\[ \text{Length of the first train} = 2x = 2 \times \frac{290}{3} = \frac{580}{3} \approx 193.33 \text{ meters} \]

### Step 6: Use the Time Taken to Cross the Platform
The first train crosses a platform in 45 seconds. The distance covered while crossing the platform is the sum of the length of the first train and the length of the platform \( p \):
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
\[ \text{Distance} = 12.5 \times 45 = 562.5 \text{ meters} \]
Thus, we have:
\[ 2x + p = 562.5 \]
Substituting \( 2x \):
\[ 193.33 + p = 562.5 \implies p = 562.5 - 193.33 \approx 369.17 \text{ meters} \]

### Final Answer
The length of the platform is approximately **369.17 meters**.

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