A person observes that the full length of a train subtends an angle of `15^@`. If the distance between the train and the person is 3km, the length of the train, calculate using the parallax method , in meters is

Options

A. 45
B. `45 pi`
C. `250 pi`
D. `75 pi`

Answer

To solve the problem of calculating the length of the train using the parallax method, we can follow these steps:

### Step 1: Understand the relationship between angle, arc length, and radius
The relationship can be expressed with the formula:
\[ L = r \cdot \theta \]
where:
- \( L \) is the length of the train (arc length),
- \( r \) is the distance from the observer to the train (radius),
- \( \theta \) is the angle subtended by the train in radians.

### Step 2: Convert the angle from degrees to radians
The angle given is \( 15^\circ \). To convert degrees to radians, we use the conversion factor:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Thus,
\[ \theta = 15^\circ \times \frac{\pi}{180} = \frac{\pi}{12} \text{ radians} \]

### Step 3: Identify the distance to the train
The distance between the observer and the train is given as \( 3 \text{ km} \). We need to convert this distance into meters:
\[ r = 3 \text{ km} = 3000 \text{ meters} \]

### Step 4: Substitute the values into the formula
Now we can substitute the values of \( r \) and \( \theta \) into the formula:
\[ L = r \cdot \theta = 3000 \cdot \frac{\pi}{12} \]

### Step 5: Calculate the length of the train
To calculate \( L \):
\[ L = 3000 \cdot \frac{\pi}{12} = \frac{3000\pi}{12} = \frac{250\pi}{1} \approx 785.4 \text{ meters} \]

### Final Result
Thus, the length of the train is approximately \( 785.4 \text{ meters} \).