https://file.aiquickdraw.com/mathbot/file/1727053426495lpsyn649.jpg

To solve the problem given in the image, we need to break down the movement of the plane into its northward and eastward components based on its bearing.

### Problem:
Calculate the distance traveled north and the distance traveled east by a plane flying on a bearing of 45° for 165 km.

### Solution:
A bearing of 45° indicates that the plane is traveling equally north and east because 45° is halfway between the north (0°) and east (90°).

1. **Components of the Distance:**
   - The northward (y) and eastward (x) components of the distance can be calculated using trigonometric functions.
   - For a bearing \(\theta = 45^\circ\):
     \[
     x = D \times \cos(45^\circ)
     \]
     \[
     y = D \times \sin(45^\circ)
     \]
   - Where \( D = 165 \, \text{km} \).

2. **Calculating the Components:**
   \[
   x = 165 \times \cos(45^\circ) = 165 \times \frac{\sqrt{2}}{2} = 165 \times 0.7071 \approx 116.7 \, \text{km}
   \]
   \[
   y = 165 \times \sin(45^\circ) = 165 \times \frac{\sqrt{2}}{2} = 165 \times 0.7071 \approx 116.7 \, \text{km}
   \]

### Result:
- **Distance traveled north:** \( \approx 116.7 \, \text{km} \)
- **Distance traveled east:** \( \approx 116.7 \, \text{km} \)

This is because at a 45° angle, the distances traveled north and east are equal.

Would you like a more detailed explanation, or have any specific questions?

### Related Questions:
1. How do you calculate the components for a plane flying at a bearing of 30°?
2. What is the formula for calculating distance components for any given bearing and distance?
3. How do you calculate the total distance traveled by a plane flying with a variable bearing?
4. How do trigonometric functions help in resolving vectors into components?
5. What would be the bearing if a plane has to travel twice the distance east compared to north?

**Tip:** Remember, when dealing with bearings, north is considered as 0° and angles are measured clockwise from the north.

Calculate the distance travelled north and the distance travelled east by a plane flying on a bearing of 45° for 165 km.

Learn how to calculate the distance traveled north and east by a plane flying on a 45° bearing for 165 km. This problem explores trigonometric concepts and uses key formulas to break down vector components.

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