A gun is fired from behind a train the driver of train hears the sound `1 (1)/(2)` min later than guard. Find the length of train if the speed of train & sound are 60 km/hr and 1100 m/min.

Options

A. 100 m
B. 150 m
C. 125 m
D. 75 m

Answer

To solve the problem, we need to find the length of the train based on the information provided about the sound of a gun fired from behind the train. Here's a step-by-step solution:

### Step 1: Convert the speeds to consistent units
The speed of the train is given as 60 km/hr. We need to convert this to meters per second (m/s):
\[ \text{Speed of train} = 60 \text{ km/hr} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = \frac{60 \times 1000}{3600} = \frac{60000}{3600} = \frac{50}{3} \text{ m/s} \]

The speed of sound is given as 1100 m/min. We convert this to meters per second:
\[ \text{Speed of sound} = 1100 \text{ m/min} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{1100}{60} = \frac{55}{3} \text{ m/s} \]

### Step 2: Understand the time difference
The driver hears the sound 1.5 minutes (or 90 seconds) later than the guard. This time difference is crucial for calculating the distance the sound travels relative to the train.

### Step 3: Calculate the distance traveled by the sound
The distance traveled by the sound in the time difference (90 seconds) can be calculated using the speed of sound:
\[ \text{Distance traveled by sound} = \text{Speed of sound} \times \text{Time} = \frac{55}{3} \text{ m/s} \times 90 \text{ s} = 1650 \text{ m} \]

### Step 4: Calculate the distance traveled by the train
During the same time (90 seconds), the train is also moving. We can calculate the distance traveled by the train:
\[ \text{Distance traveled by train} = \text{Speed of train} \times \text{Time} = \frac{50}{3} \text{ m/s} \times 90 \text{ s} = 1500 \text{ m} \]

### Step 5: Set up the equation for the length of the train
The total distance the sound travels is equal to the distance the train traveled plus the length of the train:
\[ \text{Distance traveled by sound} = \text{Distance traveled by train} + \text{Length of train} \]
Substituting the values we calculated:
\[ 1650 \text{ m} = 1500 \text{ m} + \text{Length of train} \]
\[ \text{Length of train} = 1650 \text{ m} - 1500 \text{ m} = 150 \text{ m} \]

### Final Answer
The length of the train is **150 meters**.

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