A light plane attains an airspeed of 491 km/h. The pilot sets out for a destination 785 km due north but discovers that the plane must be headed 20.5° east of due north to fly there directly. The plane arrives in 2.25 h. What was the magnitude of the wind velocity? What was direction of the wind velocity?

To find the magnitude and direction of the wind velocity, we need to break down the plane's velocity into two components: the airspeed and the wind velocity.

Given:
Airspeed of the plane (V_plane) = 491 km/h
Distance traveled (d) = 785 km
Heading angle (θ) = 20.5° east of north
Time taken (t) = 2.25 hours

Step 1: Finding the plane's ground velocity:
Since the plane is flying at an angle, we need to find the northward and eastward components of its velocity.

The northward component (V_north) can be calculated using the formula:
V_north = V_plane * cos(θ)

Substituting the values, we get:
V_north = 491 km/h * cos(20.5°)

Step 2: Finding the eastward component:
The eastward component (V_east) can be calculated using the formula:
V_east = V_plane * sin(θ)

Substituting the values, we get:
V_east = 491 km/h * sin(20.5°)

Step 3: Finding the time taken to travel north:
The time taken to travel north (t_north) can be calculated using the formula:
t_north = d / V_north

Substituting the values, we get:
t_north = 785 km / V_north

Step 4: Finding the wind velocity:
The wind velocity (V_wind) can be calculated using the formula:
V_wind = d / t_north - V_plane

Substituting the values, we get:
V_wind = 785 km / t_north - 491 km/h

Step 5: Finding the magnitude and direction of the wind velocity:
The magnitude (|V_wind|) of the wind velocity is simply the absolute value of V_wind.

The direction of the wind velocity can be determined using the northward (V_north) and eastward (V_east) components:
Direction = arctan(V_east / V_north)

Now, let's calculate the values step by step:

Step 1:
V_north = 491 km/h * cos(20.5°)
V_north ≈ 463.55 km/h

Step 2:
V_east = 491 km/h * sin(20.5°)
V_east ≈ 168.92 km/h

Step 3:
t_north = 785 km / V_north
t_north ≈ 1.694 hours

Step 4:
V_wind = 785 km / t_north - 491 km/h
V_wind ≈ 194.16 km/h

Step 5:
|V_wind| ≈ |194.16 km/h| ≈ 194.16 km/h

Direction = arctan(V_east / V_north)
Direction = arctan(168.92 km/h / 463.55 km/h)
Direction ≈ 20.5° east of north

Therefore, the magnitude of the wind velocity is approximately 194.16 km/h, and the direction of the wind velocity is approximately 20.5° east of north.