A plane leaves Seattle, flies 77.0 mi at 22.0 degrees north of east, and then changes direction to 50.0 degrees south of east. After flying at 119 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 should the crew fly to go directly to the field?
To determine the direction the rescue crew should fly to go directly to the field, we can represent the plane's displacement as a vector and then find its resultant vector.
Step 1: Determine the x and y components of the plane's displacement.
Given:
Initial displacement: 77.0 mi at 22.0 degrees north of east
Changing direction: 50.0 degrees south of east for a distance of 119 mi
The x-component of the initial displacement is calculated as:
x_initial = 77.0 mi * cos(22.0 degrees)
The y-component of the initial displacement is calculated as:
y_initial = 77.0 mi * sin(22.0 degrees)
The x-component of the displacement after changing direction is calculated as:
x_after_change = 119 mi * cos(180.0 degrees - 50.0 degrees)
The y-component of the displacement after changing direction is calculated as:
y_after_change = 119 mi * sin(180.0 degrees - 50.0 degrees)
Step 2: Sum up the x and y components of the displacements.
The total x-component is the sum of the individual x-components:
x_total = x_initial + x_after_change
The total y-component is the sum of the individual y-components:
y_total = y_initial + y_after_change
Step 3: Calculate the direction of the resultant vector.
The direction of the resultant vector can be found using trigonometry. We can calculate the angle θ using the arctan(y_total / x_total) formula.
θ = arctan(y_total / x_total)
Step 4: Calculate the magnitude of the resultant vector.
The magnitude of the resultant vector can be found using the Pythagorean theorem:
Magnitude = sqrt(x_total^2 + y_total^2)
Now, let's calculate the components and magnitude of the resultant vector to find the direction the crew should fly in.
x_initial = 77.0 mi * cos(22.0 degrees)
= 70.72 mi
y_initial = 77.0 mi * sin(22.0 degrees)
= 27.97 mi
x_after_change = 119 mi * cos(180.0 degrees - 50.0 degrees)
= -59.44 mi
y_after_change = 119 mi * sin(180.0 degrees - 50.0 degrees)
= -90.3 mi
x_total = x_initial + x_after_change
= 70.72 mi + (-59.44 mi)
= 11.28 mi
y_total = y_initial + y_after_change
= 27.97 mi + (-90.3 mi)
= -62.33 mi
θ = arctan(y_total / x_total)
= arctan((-62.33 mi) / (11.28 mi))
≈ -80.1 degrees (rounded to one decimal place)
Magnitude = sqrt(x_total^2 + y_total^2)
= sqrt((11.28 mi)^2 + (-62.33 mi)^2)
≈ 63.6 mi (rounded to one decimal place)
Therefore, the crew should fly in a direction approximately 80.1 degrees south of west to go directly to the field.
To determine the direction in which the rescue crew should fly directly to the field, we need to use vector addition.
Let's break down the problem into components.
The first leg of the journey can be represented as a vector with a magnitude of 77.0 mi and a direction of 22.0 degrees north of east. To convert this vector into its horizontal and vertical components, we use trigonometry.
The horizontal component, which represents the distance traveled in the eastward direction, can be calculated as:
Horizontal component = 77.0 mi × cos(22.0 degrees)
Similarly, the vertical component, which represents the distance traveled in the northward direction, can be calculated as:
Vertical component = 77.0 mi × sin(22.0 degrees)
Next, we consider the second leg of the journey, where the plane changes direction. This vector has a magnitude of 119 mi and a direction of 50.0 degrees south of east.
Again, we can convert this vector into its horizontal and vertical components:
Horizontal component = 119 mi × cos(180 degrees - 50.0 degrees)
Vertical component = 119 mi × sin(180 degrees - 50.0 degrees)
Now, it's time to find the net horizontal and vertical components by adding the respective components from each leg. To find the direction, we'll use the inverse tangent of the ratio of the vertical component to the horizontal component:
Direction = atan(vertical component / horizontal component)
Finally, we'll convert the direction from radians to degrees to get our answer.
So, let's calculate the direction by following these steps: