A plane leaves Seattle, flies 77.0 mi at 21.0 ∘ north of east, and then changes direction to 52.0 ∘ south of east. After flying at 124 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 to the field?
52o S. of E. = 308o CCW.
Disp. = 77[21o] + 124[308o].
Disp. = (77*Cos21+124*Cos308) + (77*sin21+124*sin308)I = 148.2 - 70.11i = 164m1[-25.3o] = 25.3o S. of E. = Direction.
The crew must fly 164 miles in a direction 25.3o S. of E.
To find the direction and distance the rescue crew should fly to reach the field, we can use vector addition.
Let's break down the flight of the plane into two vectors:
1. The first vector is the initial 77.0 mi at 21.0° north of east. We can represent this vector as V1 = 77.0 mi at 21.0°.
2. The second vector is the subsequent 124 mi at 52.0° south of east. We can represent this vector as V2 = 124 mi at -52.0° (negative angle indicates south of east).
Now, we need to find the resultant vector by adding V1 and V2.
To add vectors, we need to split them into their horizontal and vertical components and then sum them separately.
V1 can be split into its horizontal and vertical components using trigonometry:
V1 horizontal component = 77.0 mi * cos(21.0°)
V1 vertical component = 77.0 mi * sin(21.0°)
V1 horizontal component ≈ 72.29 mi
V1 vertical component ≈ 28.12 mi
Similarly, we can split V2 into horizontal and vertical components using trigonometry:
V2 horizontal component = 124 mi * cos(-52.0°)
V2 vertical component = 124 mi * sin(-52.0°)
V2 horizontal component ≈ 62.02 mi
V2 vertical component ≈ -100.48 mi
Now, we can add the horizontal and vertical components separately to find the resultant vector:
Resultant horizontal component = V1 horizontal component + V2 horizontal component
Resultant vertical component = V1 vertical component + V2 vertical component
Resultant horizontal component ≈ 72.29 mi + 62.02 mi ≈ 134.31 mi
Resultant vertical component ≈ 28.12 mi - 100.48 mi ≈ -72.36 mi
To find the magnitude and direction of the resultant vector, we can use the Pythagorean theorem and trigonometry.
Magnitude of the resultant vector:
Magnitude = sqrt((Resultant horizontal component)^2 + (Resultant vertical component)^2)
Magnitude ≈ sqrt((134.31 mi)^2 + (-72.36 mi)^2) ≈ sqrt(17999.7761 mi^2) ≈ 134.14 mi
Direction of the resultant vector:
Direction = atan2(Resultant vertical component, Resultant horizontal component)
Direction ≈ atan2(-72.36 mi, 134.31 mi) ≈ -29.4°
So, the rescue crew should fly approximately 134.14 mi at a direction of 29.4° south of east to reach the field.