Answer:
539.4 km ∠22.6°
Step-by-step explanation:
The sum of the two vectors is easily calculated by a suitable calculator. Most graphing calculators and many scientific calculators will add vectors specified in magnitude and angle form. (see attached)
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Alternate calculation
25(cos(35°), sin(35°)) +515(cos(22°), sin(22°))
≈ (20.479 +477.500, 14.339 +192.922)
= (497.978, 207.261)
The magnitude of this is ...
√(497.978^2 +207.261^2) ≈ 539.389
and the angle is ...
arctan(207.261/497.978) ≈ 22.597°
The total displacement is about 539.4 km at an angle of 22.6° to the ground.
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Comment on the problem
The magnitudes and directions given make no sense. Even if we were to assume a flat Earth, the vertical displacement relative to the starting point is more than 207 km. The Kármán line, at 100 km altitude, is the defining line between the atmosphere and outer space.
Navigation angles are rarely given relative "to the ground." Usually, they are directions measured clockwise from north. Certainly no passenger plane will ever travel at an angle of 35° to the ground. They typically change altitude at angles less than 5° to the horizontal.
