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 of finding the length of the train using the parallax method, we can follow these steps:

### Step 1: Understand the Geometry
The train subtends an angle of \(15^\circ\) at the observer's position, and the distance from the observer to the train is \(3\) km. We can visualize this situation as a circular arc where the observer is at the center of the circle.

### Step 2: Convert the Angle to Radians
To use the formula for the length of an arc, we need to convert the angle from degrees to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
For \(15^\circ\):
\[ \theta = 15 \times \frac{\pi}{180} = \frac{\pi}{12} \text{ radians} \]

### Step 3: Use the Arc Length Formula
The formula for the length of an arc \(L\) is given by:
\[ L = r \times \theta \]
where \(r\) is the radius (the distance from the observer to the train) and \(\theta\) is the angle in radians.

### Step 4: Substitute the Values
Here, \(r = 3\) km = \(3000\) meters (since we need the length in meters) and \(\theta = \frac{\pi}{12}\):
\[ L = 3000 \times \frac{\pi}{12} \]

### Step 5: Simplify the Expression
Now we can simplify the expression:
\[ L = 3000 \times \frac{\pi}{12} = \frac{3000\pi}{12} = 250\pi \text{ meters} \]

### Final Answer
Thus, the length of the train is:
\[ L = 250\pi \text{ meters} \]