Answer
The bearing the pilot should fly is $306^{\circ}$
The ground speed will be 523.8 mph
Work Step by Step
Let $a = 520~mph$
Let $b = 37~mph$
Let $c$ be the resultant of the two vectors $a$ and $b$. Note that the magnitude of $c$ is the ground speed.
Let angle $A$ be the angle that subtends side $a$. Then $A = 32^{\circ}+50^{\circ} = 82^{\circ}$
We can use the law of sines to find the angle B between the vectors a and c:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$sin~B = \frac{b~sin~A}{a}$
$B = arcsin(\frac{b~sin~A}{a})$
$B = arcsin(\frac{(37)~sin~82^{\circ}}{520})$
$B = arcsin(0.07046)$
$B = 4.0^{\circ}$
The bearing the pilot should fly is $310^{\circ}-4.0^{\circ} = 306^{\circ}$
We can find the angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}-A-B$
$C = 180^{\circ}-82^{\circ}-4.0^{\circ}$
$C = 94^{\circ}$
We can use the law of sines to find $c$:
$\frac{c}{sin~C} = \frac{a}{sin~A}$
$c = \frac{a~sin~C}{sin~A}$
$c = \frac{(520~mph)~sin~94^{\circ}}{sin~82^{\circ}}$
$c = 523.8~mph$
The ground speed will be 523.8 mph
