Answer: 34.65 miles at an angle of 325.39°
Step-by-step explanation:
Ok, the initial position is (0,0)
Then she flies 30 miles at an angle of 160°, if we count the angle counterclokwise from the x-axis, the new position will be:
p = (30*cos(120°), 30*sin(120°))
Then she travels another 15 miles at an angle of 205°, the new position is:
p = (30*cos(120°) + 15*cos(205°), 30*sin(120°) + 15*sin(205°))
p = (-28.59 , 19.64)
If she now travles X miles at an angle Y, we must have that the final position is the point (0,0)
this means that:
X*cos(Y) = -(-28.59) = 28.59
X*sin(Y) = -19.64
Now, we can find the quotient between those two equations and use that tan(x) = sin(x)/cos(x)
X*(sin(Y))/(X*cos(Y)) = -19.64/28.59
Tg(Y) = -0.69
Y = ATg(-0.69) = -34.61°
If we use only positive angles, this angle is equivalent to:
360° - 34.61° = 325.39°
now lets find the distance:
Xcos(325.39°) = 28.59
X = 28.59/cos(325.39°) = 34.65 miles.
