Help with frame and reference problem

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To determine the speed of the airplane with respect to the ground, the correct approach involves breaking down the air's velocity into its components and applying vector addition. The air's horizontal component is approximately 30.31 m/s, and the vertical component is about 17.5 m/s. The plane's speed relative to the air is 100 m/s due east, but the user incorrectly calculated its components using the wrong angle. The correct method requires using the right triangle formed by the air's velocity and the plane's velocity to find the resultant speed. The final speed of the plane with respect to the ground is approximately 116.91 m/s, confirming the need for accurate vector addition in solving the problem.
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You are on an airplane traveling due east at 100 m/s with respect to the air. The air is moving with a speed 35 m/s with respect to the ground at an angle of 30° west of due north.

What is the speed of the plane with respect to the ground?
__________

So here's what I am doing:
Drew the whole right triangle thing to bring down air into its horizontal and vertical components. Vertical turned out to be 17.5m/s and horizontal 30.31m/s.

Then I broke up the speed of the plane into its horizontal and vertical components, but I used the same angle of 30 to get horizontal speed of 86.6 and vertical speed of 50. I'm pretty sure this part is wrong.

So to get the speed of the plane wrt ground, the equation is: V[p,g]=V[p,a] + V[a,g]
so v[p,g]=86.6+30.31=116.91. That was the wrong answer. I tried subtracting-wrong answer. I even used 60 degrees instead of 30 for breaking up the plane into vertical and horizontal, but that didnt work out either. I have no idea what I am doing.
 
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is the answer about 97.34? I guess you just made the triangle wrong its a simple vector addition...
 
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Thats incorrect. Can anyone else help me with this?
 

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