A train 300m long, passed a man, walking along the line in the same direction at the rate of 3km/hr in 33 seconds. The speed of the train is

Options

A. 30 km/h
B. 32km/h
C. `32(8)/(11)` km/h
D. `35 (8)/(11)` km/h

Answer

To find the speed of the train, we can follow these steps:

### Step 1: Convert the speed of the man from km/hr to m/s
The speed of the man is given as 3 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/hr} = \frac{5}{18} \text{ m/s} \]
So,
\[ \text{Speed of the man in m/s} = 3 \times \frac{5}{18} = \frac{15}{18} = \frac{5}{6} \text{ m/s} \]

### Step 2: Calculate the relative speed of the train with respect to the man
The train and the man are moving in the same direction. Therefore, the relative speed of the train with respect to the man is:
\[ \text{Relative speed} = \text{Speed of the train} - \text{Speed of the man} \]

Let the speed of the train be \( V \) m/s. Thus,
\[ \text{Relative speed} = V - \frac{5}{6} \]

### Step 3: Use the time taken to pass the man to find the relative speed
The train passes the man in 33 seconds. The length of the train is 300 meters. Therefore, we can use the formula:
\[ \text{Distance} = \text{Relative speed} \times \text{Time} \]
Substituting the known values:
\[ 300 = \left(V - \frac{5}{6}\right) \times 33 \]

### Step 4: Solve for the speed of the train
Now, we can rearrange the equation to solve for \( V \):
\[ 300 = 33V - 33 \times \frac{5}{6} \]
Calculating \( 33 \times \frac{5}{6} \):
\[ 33 \times \frac{5}{6} = \frac{165}{6} = 27.5 \]
So, the equation becomes:
\[ 300 = 33V - 27.5 \]
Adding 27.5 to both sides:
\[ 300 + 27.5 = 33V \]
\[ 327.5 = 33V \]
Now, divide both sides by 33:
\[ V = \frac{327.5}{33} \approx 9.92 \text{ m/s} \]

### Step 5: Convert the speed of the train back to km/hr
To convert the speed from m/s back to km/hr, we use the conversion factor:
\[ 1 \text{ m/s} = 3.6 \text{ km/hr} \]
Thus,
\[ \text{Speed of the train in km/hr} = 9.92 \times 3.6 \approx 35.75 \text{ km/hr} \]

### Final Answer
The speed of the train is approximately **35.75 km/hr**.

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