A plane is flying at a speed of `720kmh^(-1)` with respect to air. The wind is blowing at a speed of `54kmh^(-1)` from west to east. With respect to ground the plane is found to be movig northwards. In which direction is the plane heading?

Options

A. North-West at angle `sin^(-1)(3/40)` to north
B. North-West at angle `tan^(-1)(3/40)` to west.
C. North-East at angle `sin^(-1)(3/40)` to north.
D. North -East at angle `tan^(-1)(3/40)` to east

Answer

To solve the problem, we need to analyze the motion of the plane and the wind vector. We will use vector addition to determine the direction in which the plane is heading.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Speed of the plane with respect to air, \( V_{pa} = 720 \, \text{km/h} \)
- Speed of the wind from west to east, \( V_w = 54 \, \text{km/h} \)
- Resultant motion of the plane with respect to the ground is northward.

2. **Draw the Vectors:**
- Represent the wind vector \( V_w \) pointing east.
- Represent the plane's velocity vector \( V_{pa} \) at an angle \( \theta \) with respect to the north direction.

3. **Set Up the Vector Components:**
- The plane's velocity can be broken down into two components:
- Northward component: \( V_{pN} = V_{pa} \cos(\theta) \)
- Eastward component: \( V_{pE} = V_{pa} \sin(\theta) \)

4. **Use the Wind's Effect:**
- Since the resultant motion of the plane is northward, the eastward component of the plane's velocity must equal the wind's velocity:
\[ V_{pE} = V_w \]
- Therefore, we have:
\[ V_{pa} \sin(\theta) = 54 \, \text{km/h} \]

5. **Substitute the Known Values:**
- Substitute \( V_{pa} = 720 \, \text{km/h} \) into the equation:
\[ 720 \sin(\theta) = 54 \]

6. **Solve for \( \sin(\theta) \):**
- Rearranging gives:
\[ \sin(\theta) = \frac{54}{720} = \frac{3}{40} \]

7. **Find the Angle \( \theta \):**
- To find \( \theta \), take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{3}{40}\right) \]

8. **Determine the Direction:**
- Since the plane is heading northwest (to counteract the eastward wind), the final heading of the plane is northwest at an angle \( \theta \) from the north.

### Final Answer:
The plane is heading northwest at an angle of \( \sin^{-1}\left(\frac{3}{40}\right) \) to the north.