A plane leaves Seattle, flies 84.0{\rm mi} at 21.0^\circ north of east, and then changes direction to 52.0^\circ south of east. After flying at 120{\rm 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? Use components to solve this problem.
How far should the crew fly to go directly to the field? Use components to solve this problem.
D = 84mi @ 21Deg. + 120mi @ -52Deg.
X = 84*cos21 + 120*cos(-52) = 152.3 Mi.
Y = 84*sin21 + 120*sin(-52) = -64.5 Mi.
tanA = Y/X = -64.5 / 152.3 = -.042343.
A = -22.9 Deg. = 22.9 Deg. S. of East.
D=X/cosA = 152.3 / cos(-22.9)=165.3 Mi.
= Dist.
a=84 mi, b=120 mi c=?
a•cos21+b•cos 55 = c•cosθ
b•sin55-a•sin21= c•sinθ
c•cosθ =84• •cos21+120•cos 55 =146.5 ...(1)
c•sinθ = 120•sin55-84•sin21 = 68.16 ...(2)
Divide (2) by (1)
tanθ =68.16/146.5 = 0.465
θ =arctan0.465 = 24.95º (south of east)
From (1)
c=146.5/cos24.95 = 161.6 mi
My mistake: I've used 55 instead of 52
To solve this problem using components, we can break down the distances and directions into their x and y components.
First, let's define our coordinate system. We will use the x-axis pointing east and the y-axis pointing north.
Given:
1. The plane flies 84.0 miles at 21.0 degrees north of east.
2. Then, it changes direction to 52.0 degrees south of east and flies 120 miles.
Step 1: Convert the distances and angles to components.
- For the first leg, the distance is 84.0 miles, and the angle is 21.0 degrees north of east. To find its x-component, we use the equation:
x_component = distance * cos(angle)
x_component = 84.0 * cos(21.0)
x_component ≈ 78.94 miles
To find its y-component, we use the equation:
y_component = distance * sin(angle)
y_component = 84.0 * sin(21.0)
y_component ≈ 30.01 miles
- For the second leg, the distance is 120 miles, and the angle is 52.0 degrees south of east. To find its x-component, we use the equation:
x_component = distance * cos(angle)
x_component = 120 * cos(180 - 52.0)
x_component ≈ -72.73 miles
To find its y-component, we use the equation:
y_component = distance * sin(angle)
y_component = 120 * sin(180 - 52.0)
y_component ≈ -92.59 miles
Step 2: Add up the x and y components.
Total x-component = 78.94 - 72.73 ≈ 6.21 miles (to the east)
Total y-component = 30.01 - 92.59 ≈ -62.58 miles (to the south)
Step 3: Find the direction and distance to go directly to the field.
To find the direction, we use the equation:
angle = tan^{-1}(y_component / x_component)
angle = tan^{-1}((-62.58) / 6.21)
angle ≈ -86.36 degrees (south of east)
To find the distance, we use the Pythagorean theorem:
distance = sqrt(x_component^2 + y_component^2)
distance = sqrt(6.21^2 + (-62.58)^2)
distance ≈ 62.79 miles
Therefore, the crew should fly approximately 62.79 miles at an angle of about 86.36 degrees south of east to go directly to the field.