DIY Solar Panels

a small aircraft is headed due south with a speed of 57.8 m/s with respect to still air. then, for 9.00 x 10^2 seconds a wind blows the plane so that it moves in a drection 45.0 degrees west of south, enen though the plane continues to point due south. the plane travels 81.0 km with respect to the ground in this time. determine the velocity (magnitude and direction) of the wind with respect to the ground. Determine the directional angle is relative to due south.

The components of the wind which changed the direction of the plane to 45 degrees west of south is the resultant of the wind direction and the airplane heading.
According to your data, I understand the solution as follows:

The airplane had a y component = 180 degrees or heading south. There is no X component to the airplane during its flight before it was subject to the wind forces

The Velocity of the airplane while the wind blowing for 900 seconds = 81000/900 = 90 m/second in the direction of 45 degrees west of south. The y component of that movement is V cos theta + 57.8 cos 45 = 0
The x component is v sin theta + 57.8 sine 45 = 90 m/s
V cos theta = 41
V sin theta = 90 -40.5 = 49.5
Tan theta = 49.5/41 =1.22 . Theta =50.6
V cos 50.6 = 41, v = 41/0.63 =65 m/s at a direction 50.6 west of south.