Check the picture representing the movement of the aircraft.
Let B be the base, A is the first location and C is the second location.
The angles of the directions S 46° E and S 55° W are as shown in the figure.
The measure of angle BAC is (90°-46°)+(90°-55°)=44°+35°=79°.
According to the cosine law, we have:
[tex]|BC|^2=|BA|^2+|AC|^2-2 \cdot|BA| \cdot |AC| \cdot cos 79^o\\\\|BC|^2=550^2+483^2-2 \cdot 550 \cdot 483 \cdot 0.19\\\\|BC|^2=302,500+233,289-100,947\\\\|BC|^2=434,842\\\\|BC|= \sqrt{434,842}\approx 659 (miles)[/tex]
Answer: 659 miles
