Please help me, I am stuck and I am getting graded on this.

Question

A plane leaves Seattle, flies 85 mi at 22° north of east, and then changes direction to 48° south of east. After flying at 115 mi in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew. In what direction and how far should the crew fly to go directly to the field?

Answer 1

This is a vector addition problem. First break each of these movements into their x-y components, keeping angles relative to a chosen axis. Lets choose East as that axis, with positive angles north of east, and negative angles south of east. 

 

X₁ = <x₁,y₁> = <85mi * cos 22, 85mi * sin (22)>

X₂ = <x2,y1> = <115mi * cos -48, 115mi * sin (-48)>

Then we do a vector addition to find the direction the plane ends up relative to the Seattle airport: 

X₃ = X₁ + X₂ = <x₃, y₃>

So add the components of X₁ and X₂ to get the components of X₃.

Then the final step is to convert from the cartesian coordinates <x₃, y₃> into circular coordinates. The distance you get from pythagorean theorem: sqrt(x₃² + y₃²), and the using a tangent: 

Tan(theta) = y₃ / x₃