Understanding the Train Crossing Problem

This problem involves calculating the length of a platform based on a train's speed, the time it takes to cross the platform, and a relationship between the train's length and the platform's length.

Step 1: Convert Speed to Consistent Units

The train's speed is given as 36 km/hr. Since the time is in seconds, we need to convert the speed to meters per second (m/sec) for consistency.

The conversion factor from km/hr to m/sec is $\frac{5}{18}$.

Speed = $36 \, \text{km/hr} \times \frac{5}{18} \, \frac{\text{m/sec}}{\text{km/hr}}$

Speed = $2 \times 5 \, \text{m/sec}$

Speed = 10 m/sec

Step 2: Define Variables and Formulate Equations

Let the length of the train be denoted by $L_T$ (in meters).

Let the length of the platform be denoted by $L_P$ (in meters).

When a train crosses a platform, the total distance the train covers is the sum of its own length and the length of the platform. The formula relating distance, speed, and time is: Distance = Speed $\times$ Time.

In this case:

$L_T + L_P = \text{Speed} \times \text{Time}$

Substituting the known values:

$L_T + L_P = 10 \, \text{m/sec} \times 30 \, \text{secs}$

$L_T + L_P = 300 \, \text{meters}$ (Equation 1)

We are also given that the length of the platform is 20% of the length of the train.

$L_P = 20\% \, \text{of} \, L_T$

$L_P = \frac{20}{100} \times L_T$

$L_P = \frac{1}{5} L_T$ (Equation 2)

Step 3: Solve for the Length of the Platform

Now we can substitute Equation 2 into Equation 1 to solve for the lengths.

Substitute $L_P = \frac{1}{5} L_T$ into $L_T + L_P = 300$:

$L_T + \frac{1}{5} L_T = 300$

To combine the terms, find a common denominator:

$\frac{5 L_T}{5} + \frac{1 L_T}{5} = 300$

$\frac{6 L_T}{5} = 300$

Now, solve for $L_T$:

$6 L_T = 300 \times 5$

$6 L_T = 1500$

$L_T = \frac{1500}{6}$

$L_T = 250 \, \text{meters}$

Now that we have the length of the train, we can find the length of the platform using Equation 2:

$L_P = \frac{1}{5} L_T$

$L_P = \frac{1}{5} \times 250 \, \text{meters}$

$L_P = 50 \, \text{meters}$

Conclusion

The length of the platform is 50 meters. This matches option 3.