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.
distance km
angle ° west of north
airport to A
Distance east: cos 30 = east / 175
east = 151.55 km
Distance north: sin 30 = north / 175 = 87.5 km
A to B
Distance north: cos 20 = north / 150
north = 140.95 km
Distance west: sin 20 = west / 150 = 51.3km
B to C =190km west
Distance west from airport to C=-151.55 + 51.3 + 190 = 89.75km
Distance north from airport = 87.5 + 140.95 = 228.45
Using Pythagoras theorem
h^2 = 89.75 ^ 2 + 228.95 ^ 2
h = 245.45 km
tan theta = 89.75/228.45
theta = 21.45 deg
Why are u using tan thetha= x component/y component
Isnt tan thetha =y component/x component??
FIND
City C is located 190 km west of the starting point. But since it's a commuter airplane, I hope it didn't take a wrong turn and end up in a completely different city!
To find the location of city C relative to the starting point, we need to determine the total horizontal and vertical displacements.
Let's break down the information given:
1. Flight from the airport to city A:
- Distance: 175 km
- Angle: 30.0° north of east
2. Flight from city A to city B:
- Distance: 150 km
- Angle: 20.0° west of north
3. Flight from city B to city C:
- Distance: 190 km
- Angle: Due west (0° west of north)
To visualize the problem, let's assume the starting point (airport) is located at the origin (0,0) on a coordinate system.
Step 1: Find the horizontal and vertical components of each flight.
- Flight from the airport to city A:
- Horizontal component: 175 km * cos(30.0°)
- Vertical component: 175 km * sin(30.0°)
- Flight from city A to city B:
- Horizontal component: 150 km * sin(20.0°)
- Vertical component: 150 km * cos(20.0°)
- Flight from city B to city C:
- Horizontal component: 190 km
- Vertical component: 0 km (since it flies due west)
Step 2: Calculate the total horizontal and vertical displacements.
- Horizontal displacement: Sum of the horizontal components of all flights
- Vertical displacement: Sum of the vertical components of all flights
Step 3: Determine the location of city C relative to the starting point.
- The x-coordinate (horizontal) of city C is the total horizontal displacement from the starting point.
- The y-coordinate (vertical) of city C is the total vertical displacement from the starting point.
So, to find the location of city C, we need to add up the horizontal and vertical displacements calculated in Step 2.
Note: The direction (north, west, etc.) in the problem may be confusing since we are dealing with angles. However, using trigonometric functions helps us compute the correct components for each direction.
Let's calculate the values:
Horizontal displacement, x-coordinate:
X = sum of horizontal components = (175 km * cos(30.0°)) + (150 km * sin(20.0°)) + 190 km
Vertical displacement, y-coordinate:
Y = sum of vertical components = (175 km * sin(30.0°)) + (150 km * cos(20.0°))
Therefore, the location of city C relative to the starting point is (X, Y) on the coordinate system.