Rather than invoking the \(\cis(\theta)\) function for complex numbers, I'd recommend standard vector notation instead. While the two are essentially mathematically equivalent, it may be confusing to students.This is an adding vectors problem.
One vector travels straight down, (220cis270).
One travels northwest, (20cis135).
See if you can add these together.
I stand corrected Otis - your diagram is indeed correct.To follow up, the diagram in post #6 doesn't show the plane arriving due south of its departure point (as stated in the exercise). Instead, it shows the plane arriving 220 miles due south of the point where the balloon (from post #4) would be at the plane's arrival time.
I drew the diagram described in post #4 like this and used the law of cosines.
View attachment 16179
My first diagram looked like this, and I used the distance formula.
View attachment 16180
(I used the Pythagorean Theorem, to determine the legs of the blue triangle.)
Expressing horizontal and vertical components of vectors, using those in some vector arithmetic and then calculating the magnitude of a resultant vector is probably the intent of the OP's assignment (they posted on the calculus board). I'll leave that demonstration for somebody else.