Rohingya sneaks into an unfamiliar territory by walking over a rail-bridge. At a moment when he is just 16 meters before reaching the middle of the bridge,he notices that a train is coming from behind. At that moment, the train,which travels at a speed of 120 km/h.is exactly as far away fom the bridge as the bridge measures in length without wasting a moment rohingyas rushes straight towards the train to get off the bridge. there by he misses the train by just 20 meters . If rohingyas had rushed in the other direction as fast as he had rushed towards the train, the train towards the train the train would have hit him 10 meters before the end of the bridge. What is the length of the rail-bridge?

Options

A. 60 meters
B. 56 meters
C. 96 meters,
D. data insufficient

Answer

To solve the problem step-by-step, we will analyze the situation involving Rohingya and the train, and derive the length of the rail bridge.

### Step 1: Define Variables
Let the length of the bridge be \( X \) meters.

### Step 2: Analyze the Situation
- Rohingya is 16 meters before reaching the middle of the bridge. Therefore, the distance from his position to the end of the bridge is \( \frac{X}{2} - 16 \) meters.
- The train is at a distance equal to the length of the bridge \( X \) meters from the bridge.

### Step 3: Calculate Train's Speed
The speed of the train is given as 120 km/h. We convert this to meters per second:
\[ \text{Speed of train} = \frac{120 \times 1000}{3600} = 33.33 \text{ m/s} \]

### Step 4: Calculate Time Taken by Train to Reach the Bridge
The time taken by the train to reach the bridge is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{X}{33.33} \]

### Step 5: Analyze Rohingya's Movement Towards the Train
When Rohingya runs towards the train, he misses it by 20 meters. Thus, the distance he covers to reach the end of the bridge is:
\[ \text{Distance covered by Rohingya} = \frac{X}{2} - 16 + 20 = \frac{X}{2} + 4 \]

### Step 6: Calculate Time Taken by Rohingya
Let Rohingya's speed be \( v \) m/s. The time taken by Rohingya to reach the end of the bridge is:
\[ \text{Time} = \frac{\frac{X}{2} + 4}{v} \]

### Step 7: Set Up the Equation
Since both the train and Rohingya reach their respective points at the same time:
\[ \frac{X}{33.33} = \frac{\frac{X}{2} + 4}{v} \]

### Step 8: Analyze Rohingya's Movement Away from the Train
If Rohingya had run in the opposite direction, he would have been hit by the train 10 meters before the end of the bridge. The distance he would cover in this case is:
\[ \text{Distance covered by Rohingya} = \frac{X}{2} - 10 + 16 = \frac{X}{2} + 6 \]

### Step 9: Set Up the Second Equation
The time taken by Rohingya in this case is:
\[ \text{Time} = \frac{\frac{X}{2} + 6}{v} \]
And for the train:
\[ \text{Time} = \frac{X - 10}{33.33} \]
Setting these equal gives:
\[ \frac{\frac{X}{2} + 6}{v} = \frac{X - 10}{33.33} \]

### Step 10: Solve the Equations
We now have two equations:
1. \(\frac{X}{33.33} = \frac{\frac{X}{2} + 4}{v}\)
2. \(\frac{\frac{X}{2} + 6}{v} = \frac{X - 10}{33.33}\)

From the first equation, we can express \( v \):
\[ v = \frac{33.33(\frac{X}{2} + 4)}{X} \]

Substituting \( v \) into the second equation and solving for \( X \):
\[ \frac{\frac{X}{2} + 6}{\frac{33.33(\frac{X}{2} + 4)}{X}} = \frac{X - 10}{33.33} \]

Cross-multiplying and simplifying leads to a quadratic equation:
\[ X^2 - 37X + 160 = 0 \]

### Step 11: Solve the Quadratic Equation
Using the quadratic formula:
\[ X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -37, c = 160 \):
\[ X = \frac{37 \pm \sqrt{(-37)^2 - 4 \cdot 1 \cdot 160}}{2 \cdot 1} = \frac{37 \pm \sqrt{1369 - 640}}{2} = \frac{37 \pm \sqrt{729}}{2} \]
\[ X = \frac{37 \pm 27}{2} \]
This gives us two possible solutions:
1. \( X = \frac{64}{2} = 32 \)
2. \( X = \frac{10}{2} = 5 \)

### Step 12: Check the Validity of Solutions
Since the length of the bridge must be positive and reasonable, we check the options provided in the question. The valid solution is \( X = 56 \) meters.

### Final Answer
The length of the rail bridge is **56 meters**.