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 3 km , the length of the train, calculated the parallax method , in meters is

Options

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

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We need to find the length of the train (L) that subtends an angle of 15 degrees at a distance of 3 km from the observer.

### Step 2: Convert the angle from degrees to radians
The angle in radians can be calculated using the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Substituting the given angle:
\[ \theta = 15^\circ \times \frac{\pi}{180} = \frac{\pi}{12} \text{ radians} \]

### Step 3: Use the formula for the length of the train
The relationship between the length of the train (L), the distance from the observer (R), and the angle subtended (\(\theta\)) is given by:
\[ L = R \cdot \theta \]
Here, R is the distance to the train, which is 3 km. We need to convert this distance into meters:
\[ R = 3 \text{ km} = 3000 \text{ m} \]

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

### Step 5: Calculate the length of the train
Now we can compute the length:
\[ L = 3000 \cdot \frac{\pi}{12} = \frac{3000\pi}{12} = 250\pi \text{ m} \]

### Step 6: Final answer
The length of the train is:
\[ L \approx 250 \cdot 3.14 \approx 785 \text{ m} \quad (\text{if you want a numerical approximation}) \]
But the exact answer in terms of \(\pi\) is:
\[ L = 250\pi \text{ m} \]