To solve this problem, let’s use the relationship between speed, distance, and time:

Formula:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Given:

1. Speed of the train = $54 \, \text{km/hr}$
2. Time to cross the platform = $45 \, \text{seconds}$
3. Length of platform = $2x \, \text{meters}$
4. Length of train = $x \, \text{meters}$

Step 1: Convert speed to meters per second.

$54 \, \text{km/hr} = 54 \times \frac{1000}{3600} = 15 \, \text{m/s}$

Step 2: Total distance covered by the train while crossing the platform.

When the train crosses the platform, the total distance it travels is the sum of the train's length and the platform's length: $\text{Distance} = x + 2x = 3x \, \text{meters}$

Step 3: Substitute values into the formula.

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ $15 = \frac{3x}{45}$

Step 4: Solve for $x$.

$15 \times 45 = 3x$ $675 = 3x$ $x = \frac{675}{3} = 225$

Final Answer:

The length of the train ($x$) is 225 meters.

---

Would you like more details or further explanations? Here are five related questions:

1. How would the solution change if the speed of the train were different?
2. What would happen if the platform length was three times the train length?
3. Can this method be applied to calculate the crossing time for another object like a car or another train?
4. How does the calculation change if the speed was given in miles per hour instead of kilometers per hour?
5. What real-life factors could affect the accuracy of these calculations (e.g., acceleration)?

Tip: Always ensure to convert units to a consistent system (e.g., meters and seconds) when solving problems involving speed, distance, and time.