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The problem involves vector addition to determine the speed and direction of an airplane flying westward relative to the air, with a wind blowing toward the southwest (45° south of west). Let's break this down:

### Given:
- Airplane speed relative to the air (\( v_p \)) = 304 mi/hr (east to west).
- Wind speed (\( w \)) = 22 mi/hr (blowing toward southwest, i.e., 45° south of west).

### Task:
1. **Find the resultant ground speed of the airplane** (vector magnitude).
2. **Determine the direction of the airplane relative to the ground**.

### Vector Analysis:
- The airplane's velocity relative to the air is directly west (negative \( x \)-axis).
- The wind velocity can be decomposed into components along the \( x \)- and \( y \)-axes:
  - The wind is blowing toward the southwest, so it makes a 45° angle with the west direction.
  - The wind's westward component (along the \( x \)-axis) is \( w_x = w \cos(45^\circ) = 22 \times \frac{\sqrt{2}}{2} \).
  - The wind's southward component (along the \( y \)-axis) is \( w_y = w \sin(45^\circ) = 22 \times \frac{\sqrt{2}}{2} \).

### Solution:
Let me compute these components and then find the resultant vector for the airplane's speed and direction relative to the ground.### Solution:
1. **The speed of the airplane relative to the ground** is approximately **288.86 mi/hr**.
2. **The direction of the airplane** is approximately **3.09° south of west**.

### Diagram:
The correct diagram would be **Option A**. In this diagram, the wind vector (\( w \)) points toward the southwest, and the airplane's velocity vector relative to the air (\( v_p \)) points westward. The resultant velocity (\( v \)) represents the airplane's velocity relative to the ground.

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Would you like further clarification on any part of this, or need more details? Here are some related questions for deeper understanding:

1. How do you break down vectors into their components?
2. Why does the wind affect the airplane's ground speed and direction?
3. How can we apply trigonometric functions to solve vector problems?
4. What would happen if the wind was blowing from a different direction?
5. How would you determine the angle of deviation for the airplane in different scenarios?

**Tip:** When solving vector problems, always decompose the vectors into their components along the axes to simplify the calculations.

An airplane flies horizontally from east to west at 304 mi/hr relative to the air. If it flies in a steady 22 mi/hr wind that blows horizontally toward the southwest (45° south of west), find the speed and direction of the airplane relative to the ground.

Solve a vector problem where an airplane flying at 304 mi/hr encounters a 22 mi/hr wind blowing toward the southwest. Learn to calculate the airplane's speed and direction relative to the ground using vector components and trigonometry.

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