A commuter airplane starts from an airport and takes the route shown in the figure below. The plane first flies to city A located 175 km away in a direction 30.0° north of east. Next, it flies for 150 km 20.0° west of north to city B. Finally, the plane flies 190 km due west, to city C. Find the location of city C relative to the location of the starting point.
You can just use basic Trig a few times.
First, find x1 and y1 (I’m defining these as the x and y components of the first triangle, the one where a is the hypotenuse. I drew them in green in the diagram above).
x1 = 175*cos(30)
y1 = 175*sin(30)
You can do the same thing for the next triangle, where 150 is the hypotenuse. I’ll call the components x2 and y2. (This triangle is orange.)
x2 = 150*sin(20)
y2= 150*cos(20)
(Note that because of the orientation of this triangle, you have the rare case where the horizontal component is sine, and the vertical component is cosine.)
Now I’m going to define a new point, let’s call it Z. This is the point between C and B where the line crosses the y-axis. If you want to find CO, you first need to find CZ and ZO. You now have everything to do this.
CZ = 190 – (x1-x2) [draw some dotted lines across the triangles to see why this is]
CZ = 190 – (175*cos(30) – 150*sin(20))
CZ = 89.75
OZ = y1 + y2
OZ = 175*sin(30) + 150*cos(20)
OZ = 228.45
Now you can find CO by using Pythagorean Theorem.
CO^2 = CZ^2 + OZ^2
CO^2 = 89.75^2 + 228.45^2
CO = 245
Lastly, you need to find the angle. You just use Trig to do this.
So the new position is 245 km away, 21.5 degrees West of North.
To find the location of city C relative to the starting point, we can break down the airplane's route into x and y components.
Step 1: Determine the x and y components of the airplane's first leg:
The airplane flies 175 km in a direction 30.0° north of east.
To find the x-component, we use the cosine function:
cos(30°) = Adjacent / Hypotenuse
cos(30°) = x / 175 km
Solving for x:
x = 175 km * cos(30°)
To find the y-component, we use the sine function:
sin(30°) = Opposite / Hypotenuse
sin(30°) = y / 175 km
Solving for y:
y = 175 km * sin(30°)
Step 2: Determine the x and y components of the airplane's second leg:
The airplane flies 150 km in a direction 20.0° west of north.
To find the x-component, we use the sine function:
sin(20°) = Adjacent / Hypotenuse
sin(20°) = x / 150 km
Solving for x:
x = 150 km * sin(20°)
To find the y-component, we use the cosine function:
cos(20°) = Opposite / Hypotenuse
cos(20°) = y / 150 km
Solving for y:
y = 150 km * cos(20°)
Step 3: Determine the x and y components of the airplane's third leg:
The airplane flies 190 km due west.
Since it travels only in the x-direction, the y-component remains zero.
Step 4: Sum up the x and y components to find the final location of city C relative to the starting point:
x_C = x1 + x2 + x3
y_C = y1 + y2 + y3
Calculate x_C:
x_C = (175 km * cos(30°)) + (150 km * sin(20°)) + (190 km * cos(180°))
Calculate y_C:
y_C = (175 km * sin(30°)) + (150 km * cos(20°)) + (0 km)
The location of city C relative to the starting point is given by the coordinates (x_C, y_C).
To find the location of City C relative to the starting point, we can break down the airplane's motion into horizontal and vertical components.
1. Horizontal Motion:
- From the starting point, the airplane flies to City A in a direction 30.0° north of east, or 60.0° east of north. This forms a right triangle, where the hypotenuse represents the distance traveled to City A and the opposite side represents the horizontal displacement (delta x) during this leg.
- We can use trigonometry to find the horizontal displacement:
delta x = distance * cos(angle)
= 175 km * cos(60.0°)
≈ 87.5 km (rounded to one decimal place)
2. Vertical Motion:
- From City A, the airplane flies for 150 km in a direction 20.0° west of north. This forms another right triangle, where the hypotenuse represents the distance traveled during this leg and the opposite side represents the vertical displacement (delta y).
- We can use trigonometry to find the vertical displacement:
delta y = distance * sin(angle)
= 150 km * sin(20.0°)
≈ 51.5 km (rounded to one decimal place)
3. Final Position:
- Finally, from City B, the airplane flies 190 km due west. This adds to the horizontal displacement and has no effect on the vertical displacement.
- The final position of City C relative to the starting point is:
x-coordinate: delta x + 190 km
= 87.5 km + 190 km
≈ 277.5 km (rounded to one decimal place)
y-coordinate: delta y
≈ 51.5 km (rounded to one decimal place)
Therefore, the location of City C relative to the starting point is approximately (277.5 km, 51.5 km).