Question
How far east do you travel if you go 2400 meters on a bearing of 150°?
Answers
GPT-4o mini
To determine how far east you travel when moving 2400 meters on a bearing of 150°, we can break the movement into its eastward (x) and northward (y) components using trigonometry.
1. **Understanding the bearing**: A bearing of 150° means you are measuring 150° clockwise from true north. In standard position:
- The east direction corresponds to 90°.
- The south direction corresponds to 180°.
Thus, a bearing of 150° puts you in the southeast quadrant.
2. **Calculating the components**:
- The angle from the east is \(150° - 90° = 60°\).
- The eastward component can be calculated using the cosine function, and the northward component can be calculated using the sine function.
3. **Calculations**:
- Eastward distance (x) = \(2400 \times \cos(150°)\)
- Northward distance (y) = \(2400 \times \sin(150°)\)
Using a calculator:
- \( \cos(150°) = -\cos(30°) = -\frac{\sqrt{3}}{2} \)
- \( \sin(150°) = \sin(30°) = \frac{1}{2} \)
Now substituting these values:
- Eastward distance (x) = \(2400 \times -\frac{\sqrt{3}}{2} \approx 2400 \times -0.866 \approx -2078.46 \) meters (since it's negative, it's traveling west).
- Northward distance (y) = \(2400 \times \frac{1}{2} = 1200 \) meters.
4. **Summary**:
The distance traveled east is approximately -2078.46 meters, which means that you actually travel about 2078.46 meters to the west as a result of the bearing of 150°.
So, while the total distance traveled is 2400 meters, the movement in the eastward direction is approximately **-2078.46 meters**, indicating a net movement westward.
1. **Understanding the bearing**: A bearing of 150° means you are measuring 150° clockwise from true north. In standard position:
- The east direction corresponds to 90°.
- The south direction corresponds to 180°.
Thus, a bearing of 150° puts you in the southeast quadrant.
2. **Calculating the components**:
- The angle from the east is \(150° - 90° = 60°\).
- The eastward component can be calculated using the cosine function, and the northward component can be calculated using the sine function.
3. **Calculations**:
- Eastward distance (x) = \(2400 \times \cos(150°)\)
- Northward distance (y) = \(2400 \times \sin(150°)\)
Using a calculator:
- \( \cos(150°) = -\cos(30°) = -\frac{\sqrt{3}}{2} \)
- \( \sin(150°) = \sin(30°) = \frac{1}{2} \)
Now substituting these values:
- Eastward distance (x) = \(2400 \times -\frac{\sqrt{3}}{2} \approx 2400 \times -0.866 \approx -2078.46 \) meters (since it's negative, it's traveling west).
- Northward distance (y) = \(2400 \times \frac{1}{2} = 1200 \) meters.
4. **Summary**:
The distance traveled east is approximately -2078.46 meters, which means that you actually travel about 2078.46 meters to the west as a result of the bearing of 150°.
So, while the total distance traveled is 2400 meters, the movement in the eastward direction is approximately **-2078.46 meters**, indicating a net movement westward.