A plane leaves Seattle, flies 84.0 at 21.0 north of east, and then changes direction to 53.0 south of east. After flying at 125 in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew .How far should the crew fly to go directly to the field?

UNITS??????

84(????),21(????)

To determine the distance the rescue crew needs to fly directly to the field, we can break down the given information into vectors and use vector addition.

Let's break down the plane's flight path into two separate vectors:

1. The initial displacement of 84.0 miles at 21.0° north of east.
2. The subsequent displacement of 125 miles at 53.0° south of east.

Now, we can add these two vectors together to find the total displacement of the plane. To do so, we can use the rules of vector addition, which involve breaking the vectors down into their x and y components.

1. For the initial displacement:
- The x-component can be calculated as 84.0 * cos(21.0°).
- The y-component can be calculated as 84.0 * sin(21.0°).

2. For the subsequent displacement:
- The x-component can be calculated as 125 * cos(-53.0°). Note that since the direction is south of east, it has a negative angle.
- The y-component can be calculated as 125 * sin(-53.0°).

Now, let's calculate the x and y components:

- Initial displacement:
- x-component = 84.0 * cos(21.0°) = 78.493
- y-component = 84.0 * sin(21.0°) = 29.210

- Subsequent displacement:
- x-component = 125 * cos(-53.0°) = 64.413
- y-component = 125 * sin(-53.0°) = -99.393

To find the total displacement, we add the x and y components together:

- Total x-component = 78.493 + 64.413 = 142.906
- Total y-component = 29.210 + (-99.393) = -70.183

Now, we can calculate the magnitude of the total displacement using the Pythagorean theorem:

Total displacement = √[(total x-component)^2 + (total y-component)^2]
= √[(142.906)^2 + (-70.183)^2]
≈ 159.8 miles

Therefore, the rescue crew should fly approximately 159.8 miles directly to the field.

Please note that the calculations are approximate due to rounding.