The length of a train and that of a platform are equal . If with the speed of 54 km/hr , the train crosses the platform in ` 1 (1)/(2)` minutes , then what is the length of the platform in metres?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Problem
The length of the train and the platform are equal. Let's denote the length of both the train and the platform as \( x \) meters.

### Step 2: Calculate the Total Distance Covered
When the train crosses the platform, it covers a distance equal to the length of the train plus the length of the platform. Therefore, the total distance covered by the train is:
\[ \text{Distance} = x + x = 2x \text{ meters} \]

### Step 3: Convert Speed from km/hr to m/s
The speed of the train is given as 54 km/hr. To convert this speed into meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]
So,
\[ \text{Speed} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 4: Convert Time from Minutes to Seconds
The time taken to cross the platform is given as \( 1 \frac{1}{2} \) minutes. We convert this into seconds:
\[ 1 \frac{1}{2} \text{ minutes} = 1 \times 60 + 30 = 90 \text{ seconds} \]

### Step 5: Use the Formula for Distance
We know that distance is equal to speed multiplied by time. Therefore, we can set up the equation:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the values we have:
\[ 2x = 15 \text{ m/s} \times 90 \text{ s} \]

### Step 6: Calculate the Total Distance
Calculating the right side:
\[ 2x = 15 \times 90 = 1350 \text{ meters} \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{1350}{2} = 675 \text{ meters} \]

### Conclusion
The length of the platform (which is equal to the length of the train) is:
\[ \text{Length of the platform} = 675 \text{ meters} \]