An aircraft flies due North from an airport. After 800 Kilometers, it then travels on a bearing of 050° for 500 Kilometers, at which point it crosses a village. Calculate the distance and bearing of the village from the airport?
translating the angles to standard math and trig notation
North ----> 90°
bearing of 050 ----> 40°
r = (800cos90,800sin90) + (500cos40,500sin40)
= (0, 800) + (383.022, 321.394)
= (383.022, 1121.394)
|r| = √(383.022^2 + 1121.394^2)
= 1185.00 km
tan angle = 1183.0222/321.394 = 3.681
angle = 71.1°
so your bearing would be 18.9°
typo:
3rd last line should say:
tan angle = 1121.394/383.022 = 2.92775..
(the rest is correct)
To calculate the distance and bearing of the village from the airport, we can use the concept of vector addition. Let's break down the two legs of the flight and then combine them:
1. Leg 1: Due North for 800 kilometers
- This leg is a straight line due North, so we can consider it as movement only in the North direction (0°) and no movement in the East direction.
- Therefore, the coordinates for this leg would be (0, 800).
2. Leg 2: Bearing of 050° for 500 kilometers
- The bearing of 050° means the aircraft is moving 50° clockwise from North.
- We can break this leg into its North (N) and East (E) components using trigonometry.
- The North component can be calculated as: N = 500 * cos(50°)
- And the East component is: E = 500 * sin(50°)
- Therefore, the coordinates for this leg would be (E, N).
Now, let's calculate the coordinates for the village by combining the two legs:
Village coordinates = (Leg 1 coordinates) + (Leg 2 coordinates)
Village coordinates = (0, 800) + (E, N)
Substituting the values:
Village coordinates = (0, 800) + (500 * sin(50°), 500 * cos(50°))
Village coordinates = (500 * sin(50°), 800 + 500 * cos(50°))
Using a calculator:
Village coordinates ≈ (384.28, 1073.56)
To calculate the distance between the airport and the village, we can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the airport (0, 0) and the village (384.28, 1073.56):
Distance = √((384.28 - 0)^2 + (1073.56 - 0)^2)
Distance ≈ √(147626.1984 + 1153935.9936)
Distance ≈ √1301562.192
Distance ≈ 1140.9 kilometers (rounded to one decimal place)
To calculate the bearing of the village from the airport, we can use trigonometry. The bearing is the angle between the North direction and the line connecting the airport and the village.
Bearing = arctan((y2 - y1) / (x2 - x1))
Using the coordinates of the airport (0, 0) and the village (384.28, 1073.56):
Bearing = arctan((1073.56 - 0) / (384.28 - 0))
Bearing ≈ arctan(1073.56 / 384.28)
Bearing ≈ arctan(2.794)
Bearing ≈ 70.2° (rounded to one decimal place)
Therefore, the distance of the village from the airport is approximately 1140.9 kilometers, and the bearing of the village from the airport is approximately 70.2°.
To calculate the distance and bearing of the village from the airport, we can use the concept of vectors and trigonometry.
Step 1: Visualize the situation
Imagine a map with an airport as the starting point of the aircraft. The aircraft travels due North for 800 kilometers and then changes direction to 050° for an additional 500 kilometers until it reaches the village.
Step 2: Find the components of the aircraft's displacement
The displacement of the aircraft consists of two components - one in the northerly direction and the other in the easterly direction. Let's calculate these components.
The north component: Since the aircraft traveled due North for 800 kilometers, the north component of its displacement is simply 800 kilometers.
The east component: To find the east component, we use trigonometry. The angle between the aircraft's bearing and the easterly direction is 90° - 50° = 40°. We can now use the cosine function to find the east component.
east component = 500 kilometers * cos(40°)
Step 3: Calculate the total displacement
Now that we have both the north and east components, we can use the Pythagorean theorem to find the total displacement from the airport to the village.
total displacement = √(north component^2 + east component^2)
Step 4: Calculate the bearing
To find the bearing, we can use the inverse tangent function. The bearing is the angle formed between the easterly direction and the line connecting the airport to the village.
bearing = arctan(east component / north component)
Let's plug in the values and calculate the result:
north component = 800 kilometers
east component = 500 kilometers * cos(40°)
total displacement = √(800^2 + (500 * cos(40°))^2)
bearing = arctan((500 * cos(40°)) / 800)
Using a calculator, the total displacement is approximately 981.10 kilometers, and the bearing is approximately 31.98°.
Therefore, the distance of the village from the airport is approximately 981.10 kilometers, and the bearing is approximately 31.98°.