A train crosses a person in 24 sec. but it crosses a 126 m long platform in 33 sec. Find the length of train, also find the speed of train.

Options

A. 336 m and 50.4 km/hr
B. 316 m and 55.4 km/hr
C. 296m and 60.8 km/hr
D. 364 m and 50.4 km/hr

Answer

To solve the problem, we need to find the length of the train and its speed based on the information given. Here's the step-by-step solution:

### Step 1: Define Variables
Let:
- \( L \) = Length of the train (in meters)
- \( S \) = Speed of the train (in meters per second)

### Step 2: Use the Information Given
1. The train crosses a person in 24 seconds.
- When the train crosses a person, it covers a distance equal to its own length.
- Therefore, we can write the equation:
\[ L = S \times 24 \quad \text{(1)} \]

2. The train crosses a 126 m long platform in 33 seconds.
- When the train crosses the platform, it covers a distance equal to the length of the train plus the length of the platform.
- Therefore, we can write the equation:
\[ L + 126 = S \times 33 \quad \text{(2)} \]

### Step 3: Substitute Equation (1) into Equation (2)
From Equation (1), we have \( S = \frac{L}{24} \). Substitute this into Equation (2):
\[ L + 126 = \left(\frac{L}{24}\right) \times 33 \]

### Step 4: Simplify the Equation
Multiply both sides by 24 to eliminate the fraction:
\[ 24(L + 126) = 33L \]
Expanding the left side:
\[ 24L + 3024 = 33L \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ 33L - 24L = 3024 \]
\[ 9L = 3024 \]

### Step 6: Solve for \( L \)
Dividing both sides by 9:
\[ L = \frac{3024}{9} = 336 \text{ meters} \]

### Step 7: Find the Speed \( S \)
Now, substitute \( L \) back into Equation (1) to find \( S \):
\[ S = \frac{L}{24} = \frac{336}{24} = 14 \text{ m/s} \]

### Step 8: Convert Speed to km/h
To convert the speed from meters per second to kilometers per hour, multiply by \( \frac{18}{5} \):
\[ S = 14 \times \frac{18}{5} = 50.4 \text{ km/h} \]

### Final Answers
- Length of the train: **336 meters**
- Speed of the train: **50.4 km/h**

---