ANSWER:
75.8 miles per hour, 83.3°
STEP-BY-STEP EXPLANATION:
Given:
Plane velocity is 350 mi/hr N70E° (90 - 70 = 20°)
Ground velocity is 390 mi/hr, 30°
The ground velocity is the magnitude V + W, just like this:
[tex]\begin{gathered} V+W=390\cdot\cos 30i+390\cdot\sin 30j \\ V+W=337.7i+195j \end{gathered}[/tex]
We know that for V it would be:
[tex]\begin{gathered} V=350\cdot\cos 20i+350\cdot\sin 20j \\ V=328.9i+119.7j \end{gathered}[/tex]
Therefore:
[tex]\begin{gathered} V+W-V=337.7i+195j-328.9i-119.7j \\ W=8.8i+75.3j \end{gathered}[/tex]
We calculate the magnitude using the norm of the vector, like this:
[tex]\begin{gathered} |W|=\sqrt{\left(8.8\right)^2+\left(75.3\right)^2} \\ |W|=\sqrt[]{77.4+5670.9} \\ |W|=\sqrt[]{5747.53} \\ |W|=75.8 \end{gathered}[/tex]
The speed of wind is 75.8 miles per hour
We calculate the direction of the angle as follows:
[tex]\begin{gathered} \tan \theta=\frac{y}{x} \\ \theta=\tan ^{-1}\mleft(\frac{75.3}{8.8}\mright) \\ \theta=83.3\degree \end{gathered}[/tex]
The direction angle of the wind is 83.3°
