An express train, Train X, traveling at a uniform speed of 108 km/h, takes 30 seconds to completely cross a station platform that is 400 meters long. Shortly after passing the platform, Train X meets Train Y running in the opposite direction at a speed of 54 km/h. If the two trains cross each other completely in 24 seconds, what is the length of Train Y in meters?

The correct answer is 580.

Step 1: Convert the speeds from km/h to m/s

To convert speed from km/h to m/s, multiply by $\frac{5}{18}$.

Speed of Train X:

$\text{Speed of Train X}=108\times \frac{5}{18}=30\text{m/s}$

Speed of Train Y:

$\text{Speed of Train Y}=54\times \frac{5}{18}=15\text{m/s}$

Step 2: Determine the length of Train X

When a train completely crosses a platform, the total distance covered is equal to the sum of the length of the train and the length of the platform.

$\text{Total Distance}=\text{Speed of Train X}\times \text{Time}$

Let $L_X$ be the length of Train X. Given that the platform length is 400 meters and the crossing time is 30 seconds:

$L_X+400=30\times 30$

$L_X+400=900$

$L_X=900-400=500\text{meters}$

Step 3: Calculate the relative speed of the two trains

Because Train X and Train Y are running in opposite directions, their relative speed is the sum of their individual speeds:

$\text{Relative Speed}=30+15=45\text{m/s}$

Step 4: Calculate the length of Train Y

When two trains cross each other completely, the total distance covered equals the sum of the lengths of both trains.

Let $L_Y$ be the length of Train Y. The two trains cross in 24 seconds:

$\text{Total Distance}=\text{Relative Speed}\times \text{Time}$

$L_X+L_Y=45\times 24$

$500+L_Y=1080$

$L_Y=1080-500=580\text{meters}$

Therefore, the length of Train Y is 580 meters.