Answer
The speed of the plane relative to the air is $~~236~km/h$
Work Step by Step
Let $v_{pg}$ be the velocity of the plane relative to the ground.
Let $v_{pa}$ be the velocity of the plane relative to the air.
Let $v_{ag}$ be the velocity of the air relative to the ground.
We can find the east-west component of the velocity of the plane relative to the air:
$v_{pg,x} = v_{pa,x}+v_{ag,x}$
$v_{pa,x} = v_{pg,x} - v_{ag,x}$
$v_{pa,x} = 0 - (42~km/h)~cos~20^{\circ}~(east)$
$v_{pa,x} = -39.5~km/h~(east)$
$v_{pa,x} = 39.5~km/h~(west)$
We can find the north-south component of the velocity of the plane relative to the air:
$v_{pg,y} = v_{pa,y}+v_{ag,y}$
$v_{pa,y} = v_{pg,y} - v_{ag,y}$
$v_{pa,y} = (220~km/h) - (-42~km/h)~sin~20^{\circ}$
$v_{pa,y} = (220~km/h) + (14.4~km/h)$
$v_{pa,y} = 234.4~km/h ~(north)$
We can find the speed of the plane relative to the air:
$v_{pa} = \sqrt{(29.5~km/h)^2+(234.4~km/h)^2}$
$v_{pa} = 236~km/h$
The speed of the plane relative to the air is $~~236~km/h$
