We can solve this problem using the concept of relative speed and the formula for distance:

Step 1: Calculate the relative speed

Since the two trains are moving in opposite directions, we add their speeds to get the relative speed:

$\text{Relative speed} = 108 \, \text{km/h} + 90 \, \text{km/h} = 198 \, \text{km/h}$

Step 2: Convert relative speed into meters per second

To work with time in seconds, we need to convert the speed from km/h to m/s. The conversion factor is $\frac{5}{18}$:

$198 \, \text{km/h} = 198 \times \frac{5}{18} = 55 \, \text{m/s}$

Step 3: Use the time and speed to find the total distance

The total distance covered in 12 seconds is:

$\text{Total distance} = \text{Relative speed} \times \text{Time} = 55 \, \text{m/s} \times 12 \, \text{s} = 660 \, \text{meters}$

Step 4: Calculate the length of the other train

The total distance covered is the sum of the lengths of both trains. We know that the length of the first train is 360 meters. So the length of the other train is:

$\text{Length of the other train} = 660 \, \text{m} - 360 \, \text{m} = 300 \, \text{m}$

Conclusion

The length of the other train is 300 meters.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the formula change if the trains were moving in the same direction?
2. What would happen if the speed of the first train increased by 10 km/h?
3. How would the result change if the second train's speed was reduced to 60 km/h?
4. Can we solve this problem if the crossing time is unknown but both train lengths are known?
5. What role does the relative speed play in real-life train crossing scenarios?

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