Question 1204112
**1. Define the Coordinate System**

* Let's define our coordinate system with:
    * **East as the positive x-axis**
    * **North as the positive y-axis**

**2. Resolve the Velocities into Components**

* **Plane's Velocity:**
    * Magnitude: 340 km/hr
    * Direction: 12° North of East 
        * x-component: 340 * cos(12°) = 332.54 km/hr (East)
        * y-component: 340 * sin(12°) = 70.43 km/hr (North)

* **Wind Velocity:**
    * Magnitude: 40 km/hr
    * Direction: 34° South of East 
        * x-component: 40 * cos(34°) = 33.14 km/hr (East)
        * y-component: -40 * sin(34°) = -22.36 km/hr (South)

**3. Calculate the Resultant Velocity**

* **Resultant x-component:** 332.54 km/hr (plane) + 33.14 km/hr (wind) = 365.68 km/hr
* **Resultant y-component:** 70.43 km/hr (plane) - 22.36 km/hr (wind) = 48.07 km/hr

**4. Calculate the Magnitude of the Resultant Velocity**

* Magnitude = √(x-component² + y-component²) 
* Magnitude = √(365.68² + 48.07²) 
* Magnitude ≈ 369.45 km/hr

**5. Calculate the Direction of the Resultant Velocity**

* Direction = arctan(y-component / x-component)
* Direction = arctan(48.07 / 365.68) 
* Direction ≈ 7.48° North of East

**Therefore:**

* **Magnitude of the plane's actual velocity:** 369.45 km/hr
* **Direction of the plane's actual velocity:** 7.48° North of East

**In summary:**

The plane's actual velocity is 369.45 km/hr at a direction of 7.48° North of East, taking into account the effect of the wind.