Question

An airplane travels 1000 miles on a bearing of 210°. then, it changes to a bearing 270° and travels 500 miles. how far is the airplane from the starting point?

Answer 1

$\text{Let the Airplane start from the point S and travels 1000 miles on a bearing of}\\ \\ 210^{\circ} \text{ to the point F. and then travels 500 milese at a bearing of }270^{\circ} \text{ and reach L}.\\ \\ \text{so in the triangle SFL, }\angle F=90^{\circ}+30^{\circ}=120^{\circ}\\ \\ \text{so in the triangle SFL, using the Law of Cosine, we get}\\ \\ (SL)^2=(LF)^2+(SF)^2-2(LF)(SF)\cos F$

$\Rightarrow (SL)^2=(500)^2+(1000)^2-2(500)(1000)\cos(120^{\circ})\\ \\ \Rightarrow SL=\sqrt{(500)^2+(1000)^2-2(500)(1000)\cos(120^{\circ})}\\ \\ \Rightarrow SL=1323$

So the distance of the airplane from the starting point is: 1323 miles