A plane leaves Seattle, flies 85 mi at 22 degrees north of east, and then changes direction to 48 degrees south of east. After flying 115 mi in this new direction, the pilot must make an emergency landing on a field. The Seattle airport facility dispatches a rescue crew.
a) In what direction and how far should the crew fly to go directly to the field? Use the method of components to solve.
To find the direction and distance the rescue crew should fly to go directly to the field, we can break down the given distances and directions into their respective components.
Let's consider the initial leg of the plane's flight:
- The plane flies 85 miles at 22 degrees north of east.
To break this down into components:
- The component of the distance in the east direction is given by: 85 miles * cos(22 degrees).
- The component of the distance in the north direction is given by: 85 miles * sin(22 degrees).
Now, let's consider the second leg of the plane's flight after changing direction:
- The plane flies 115 miles in a direction 48 degrees south of east.
To break this down into components:
- The component of the distance in the east direction is given by: 115 miles * cos(48 degrees).
- The component of the distance in the north direction is given by: -115 miles * sin(48 degrees) [negative because it's south of east].
Now, we can calculate the total east and north components by summing the components of each leg:
East component = (85 miles * cos(22 degrees)) + (115 miles * cos(48 degrees))
North component = (85 miles * sin(22 degrees)) - (115 miles * sin(48 degrees))
To find the direction and distance the rescue crew should fly, we can use the components obtained above:
Distance = sqrt((East component)^2 + (North component)^2)
Direction = arctan(North component / East component)
Plugging in the values and performing the calculations will give us the desired answers.