Answer:
(a) north: 172.5 mph; west: 298.8 mph
(b) 363.9 mph at 298.9°
Explanation:
The components of velocity are found using the formula for polar to rectangular coordinate conversion. The actual ground speed will be the result of adding the velocity vectors of the airplane and wind.
__
(a)
The usual formula for converting polar coordinates (r; θ) to rectangular coordinates (x, y) is ...
(x, y) = r(cos(θ), sin(θ))
When this formula is used with bearing angles, the resulting components are (north, east).
The airplane velocity components are ...
north: 345·cos(300°) = 172.5 . . . . mph
east: 345·sin(300°) ≈ -298.8 . . . . mph (or 298.8 mph to the west)
__
(b)
The sum of the two velocity vectors is found by a calculator to be ...
345∠300° +20∠280° = 363.9∠298.9°
If you wanted to calculate this by hand, you would find the components of the wind speed, add them to the components of the plane speed, and convert the rectangular components back to polar coordinates. The usual way that is done is using the formula ...
r = √(x² +y²)
θ = arctan(y/x) . . . . paying attention to quadrant
If you use x=northerly component, y=easterly component, then the resulting angle is the bearing angle.
_____
Additional comment
Some calculators are able to convert directly between (x, y) coordinates and (r; θ) coordinates. The calculator shown in the attachment converts (r; θ) coordinates to/from complex numbers. For the purpose here, the real part of the number is the northerly component, and the imaginary part is the easterly component.
The conversion to a complex number is x +iy = r(cos(θ) +i·sin(θ)), sometimes abbreviated r·cis(θ). The notation used here is even more compact: r∠θ. Conveniently, the calculator we used is able to employ this notation.