Relative Velocity Airplane Problem - Air and Ground speed

[Rock32]

Homework Statement

A jet flies with an airspeed of 310 mph due east (assuming +x is due east). It is climbing vertically with a velocity of 50 mph. A wind blowing toward northeast has a speed of 55 mph.

Homework Equations

Find the x,y,z components of the plane's velocity relative to the ground.

The Attempt at a Solution

Since nothing is affecting the z direction except the plane climbing, the z component would just be 50 mph.

Because the wind is blowing toward the northeast, this adds a y component of
(55 mph)*sin(45) =38.9 mph

Both those answers are correct, but I am stuck on the x component.

I tried adding 55mph*cos(45) to the 310 mph airspeed (since the plane is flying due east) to get 350 mph, but that is wrong.

I also tried using the airspeed as the final vector's magnitude (i.e. x^2 + y^2 + z^2 = 310) using the correct answers above for y and z, but that is also wrong.

Any help would be great.

 

[rl.bhat]

What is the exact value of x component of the velocity?