Answer :
Sure! Let's solve this step-by-step:
1. Identify and note the given data:
- The aircraft flew the first leg of 80 km on a bearing of 42°.
- The aircraft flew the second leg of 120 km on a bearing of 140°.
2. Convert the bearings into Cartesian coordinates:
- Let's define the origin as the starting point `(0,0)`.
- We will break down the flight paths into their north-south (y) and east-west (x) components.
3. Calculate the coordinates after the first leg:
- The bearing of 42° means we need to convert this polar coordinate to Cartesian coordinates.
- We use trigonometry:
- `x1 = distance1 cos(bearing1)`
- `y1 = distance1 sin(bearing1)`
- Here, `distance1 = 80 km` and `bearing1 = 42°`.
4. Calculate the coordinates after the second leg:
- The new position will be an addition of the next leg to the previous coordinates:
- `x2 = x1 + distance2 cos(bearing2)`
- `y2 = y1 + distance2 sin(bearing2)`
- Here, `distance2 = 120 km` and `bearing2 = 140°`.
5. Calculate the overall distance from the starting point:
- Use the Pythagorean theorem to find the distance from the origin to the final coordinates `(x2, y2)`:
- `distance_from_start = sqrt(x2^2 + y2^2)`
6. Calculate the final bearing from the starting point:
- The bearing is the angle from the north in a clockwise direction.
- Use the `atan2` function to find the angle:
- `bearing_from_start = atan2(y2, x2)`
- Convert this angle from radians to degrees.
7. Adjust the bearing to ensure it is in the range [0, 360) degrees:
- If the bearing is negative, add 360° to it.
Based on the steps outlined, after inputting the values and performing the calculations, we find:
- Distance from the starting point: 134.64 km (rounded to two decimal places).
- Bearing from the starting point: 103.96° (rounded to two decimal places).
So, the aircraft is 134.64 km away from the starting point, and the bearing of the aircraft from the starting point is 103.96°.
1. Identify and note the given data:
- The aircraft flew the first leg of 80 km on a bearing of 42°.
- The aircraft flew the second leg of 120 km on a bearing of 140°.
2. Convert the bearings into Cartesian coordinates:
- Let's define the origin as the starting point `(0,0)`.
- We will break down the flight paths into their north-south (y) and east-west (x) components.
3. Calculate the coordinates after the first leg:
- The bearing of 42° means we need to convert this polar coordinate to Cartesian coordinates.
- We use trigonometry:
- `x1 = distance1 cos(bearing1)`
- `y1 = distance1 sin(bearing1)`
- Here, `distance1 = 80 km` and `bearing1 = 42°`.
4. Calculate the coordinates after the second leg:
- The new position will be an addition of the next leg to the previous coordinates:
- `x2 = x1 + distance2 cos(bearing2)`
- `y2 = y1 + distance2 sin(bearing2)`
- Here, `distance2 = 120 km` and `bearing2 = 140°`.
5. Calculate the overall distance from the starting point:
- Use the Pythagorean theorem to find the distance from the origin to the final coordinates `(x2, y2)`:
- `distance_from_start = sqrt(x2^2 + y2^2)`
6. Calculate the final bearing from the starting point:
- The bearing is the angle from the north in a clockwise direction.
- Use the `atan2` function to find the angle:
- `bearing_from_start = atan2(y2, x2)`
- Convert this angle from radians to degrees.
7. Adjust the bearing to ensure it is in the range [0, 360) degrees:
- If the bearing is negative, add 360° to it.
Based on the steps outlined, after inputting the values and performing the calculations, we find:
- Distance from the starting point: 134.64 km (rounded to two decimal places).
- Bearing from the starting point: 103.96° (rounded to two decimal places).
So, the aircraft is 134.64 km away from the starting point, and the bearing of the aircraft from the starting point is 103.96°.
