An aero plane has to go from point $A$ to another point $B, 500 \mathrm{~km}$ away due to $30^{\circ}$ east of north. A wind is blowing due north at a seed of $20 \mathrm{~m} / \mathrm{s}$. The air speed of the plane is $150 \mathrm{~m} / \mathrm{s}$.
(a) Find the direction in which the pilot should head the plane to reach the point $B$.
(b) Find the time taken by the plane to gr from $A$ to $B$.
(a)
In $\triangle \mathrm{ACB}$
Using sin formula
$\frac{20}{\sin \phi}=\frac{150}{\sin 30^{\circ}}$
$\sin \emptyset=\frac{1}{15}$
$\emptyset=\sin ^{-1}\left(\frac{1}{15}\right)$ east of the line $A B$
(b)
$\emptyset=3^{\circ} 48^{\prime}$. Angle between two vector $=30+3^{\circ} 48^{\prime}$
$R=\sqrt{A^{2}+B^{2}+2 A B \cos \theta}$
$\mathrm{R}=167 \mathrm{~m} / \mathrm{s}$
Time $=\frac{\text { distance }}{\text { speed }}=\frac{500 \times 10^{3}}{167}$
$=2994 \mathrm{sec}$
$\mathrm{T}=60 \approx 50 \mathrm{~min}$
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