Shatabdhi express travelling at 168 kmph and overtook Rajdhani express, traveling in the same direction in 20 seconds. If Rajdhani express had been traveling at twice its speed, then Shatabdhi express would have taken 45 seconds to overtake it. Find the length of Shatabdi express, given that its length is twice of the length of Rajdhani express?

Options

A. 180 m
B. 720 m
C. 360 m
D. 400 m

Answer

To solve the problem step by step, we need to find the length of the Shatabdhi Express given that its length is twice that of the Rajdhani Express. Let's denote the length of the Rajdhani Express as \( x \) meters, which makes the length of the Shatabdhi Express \( 2x \) meters.

### Step 1: Define Variables
- Let the length of the Rajdhani Express be \( x \) meters.
- Therefore, the length of the Shatabdhi Express is \( 2x \) meters.
- Let the speed of the Rajdhani Express be \( y \) km/h.
- The speed of the Shatabdhi Express is given as 168 km/h.

### Step 2: Set Up the First Equation
When the Shatabdhi Express overtakes the Rajdhani Express in 20 seconds, the distance covered is the sum of their lengths:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
The relative speed when both trains are moving in the same direction is:
\[ \text{Relative Speed} = 168 - y \text{ km/h} \]
Converting this speed to meters per second:
\[ \text{Relative Speed} = \frac{(168 - y) \times 1000}{3600} = \frac{(168 - y) \times 5}{18} \text{ m/s} \]
Thus, the equation for the first condition becomes:
\[ 2x + x = \left(\frac{(168 - y) \times 5}{18}\right) \times 20 \]
This simplifies to:
\[ 3x = \frac{(168 - y) \times 100}{18} \]

### Step 3: Set Up the Second Equation
If the Rajdhani Express had been traveling at twice its speed, then its speed would be \( 2y \) km/h. The time taken to overtake in this case is 45 seconds:
\[ 3x = \left(\frac{(168 - 2y) \times 5}{18}\right) \times 45 \]
This simplifies to:
\[ 3x = \frac{(168 - 2y) \times 375}{18} \]

### Step 4: Equate the Two Equations
Now we have two equations:
1. \( 3x = \frac{(168 - y) \times 100}{18} \)
2. \( 3x = \frac{(168 - 2y) \times 375}{18} \)

Setting them equal to each other:
\[ \frac{(168 - y) \times 100}{18} = \frac{(168 - 2y) \times 375}{18} \]
Cancelling out \( \frac{1}{18} \) from both sides:
\[ (168 - y) \times 100 = (168 - 2y) \times 375 \]

### Step 5: Solve for \( y \)
Expanding both sides:
\[ 16800 - 100y = 63000 - 750y \]
Rearranging gives:
\[ 750y - 100y = 63000 - 16800 \]
\[ 650y = 46200 \]
\[ y = \frac{46200}{650} = 71.08 \text{ km/h} \]

### Step 6: Find \( x \)
Now substituting \( y \) back into one of the equations to find \( x \):
Using the first equation:
\[ 3x = \frac{(168 - 71.08) \times 100}{18} \]
Calculating:
\[ 3x = \frac{96.92 \times 100}{18} = \frac{9692}{18} \approx 538.44 \]
Thus:
\[ x \approx \frac{538.44}{3} \approx 179.48 \text{ meters} \]

### Step 7: Find Length of Shatabdhi Express
The length of the Shatabdhi Express is:
\[ 2x \approx 2 \times 179.48 \approx 358.96 \text{ meters} \]
Rounding gives us approximately \( 360 \text{ meters} \).

### Final Answer
The length of the Shatabdhi Express is approximately **360 meters**.