A plane is flying on a bearing of 260 degrees at 430 mph. It's velocity in component form is -423.467,-74.669. At the same time, there is a wind blowing at a bearing of 30 degrees at 60mph. Find the ground speed of the plane.

To find the ground speed of the plane, we first need to determine the velocity components of the wind and then combine them with the velocity components of the plane.

1. **Convert the wind's bearing to standard position**: A bearing of 30 degrees means the angle is measured clockwise from the north. In standard position, this corresponds to an angle of \( 30^\circ \) (i.e., the same angle).

2. **Calculate the wind's velocity components**:
   \[
   V_{\text{wind}, x} = 60 \cdot \cos(30^\circ) = 60 \cdot \frac{\sqrt{3}}{2} \approx 51.961
   \]
   \[
   V_{\text{wind}, y} = 60 \cdot \sin(30^\circ) = 60 \cdot \frac{1}{2} = 30
   \]
   Thus, the wind's velocity in component form is approximately \( (51.961, 30) \).

3. **Add the plane's velocity components to the wind's components**:
   \[
   V_{\text{ground}, x} = -423.467 + 51.961 \approx -371.506
   \]
   \[
   V_{\text{ground}, y} = -74.669 + 30 \approx -44.669
   \]

4. **Calculate the ground speed (magnitude of the ground velocity)**:
   \[
   \text{Ground speed} = \sqrt{V_{\text{ground}, x}^2 + V_{\text{ground}, y}^2}
   \]
   \[
   = \sqrt{(-371.506)^2 + (-44.669)^2} \approx \sqrt{138,205.34 + 1,993.42} \approx \sqrt{140,198.76} \approx 374.45 \text{ mph}
   \]

Therefore, the ground speed of the plane is approximately **374.45 mph**. 

Would you like more details on any specific part of the calculations, or do you have any questions? Here are some related questions to consider:

1. How do you convert bearings to standard angles?
2. What are the implications of wind on flight paths?
3. How do you calculate velocity components for different angles?
4. Why is it important to find ground speed in aviation?
5. How would changes in wind speed affect the ground speed? 

**Tip**: Always ensure to convert angles correctly when dealing with bearings and standard position.

Explore the calculation of ground speed for a plane flying at a bearing of 260 degrees with a wind at 30 degrees. This problem illustrates vector addition and the use of trigonometry to determine ground speed, resulting in approximately 374.45 mph.

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