An airplane is flying at an airspeed of 510 km/hr in a wind blowing at 60 km/hr toward the southeast. In what direction should the plane head to end up going due east? (Round your answer to two decimal places.)
  °  ---Select--- north east south west north of east north of west south of east south of west 

What is the airplane's speed relative to the ground? (Round your answer to the nearest whole number.)
  km/hr

To solve this problem, we need to use vector addition to find both the required direction for the airplane and its speed relative to the ground.

### Step 1: Break Down the Wind's Velocity
The wind is blowing at 60 km/hr toward the southeast. Since southeast is at a 45° angle between south and east, we can break the wind's velocity into its eastward and southward components:
- Eastward component: \( 60 \times \cos(45^\circ) = 60 \times \frac{1}{\sqrt{2}} = 42.43 \) km/hr
- Southward component: \( 60 \times \sin(45^\circ) = 60 \times \frac{1}{\sqrt{2}} = 42.43 \) km/hr

### Step 2: Set Up the Airplane's Velocity Relative to the Ground
Let the airplane's heading angle be \( \theta \) north of east. The airplane's airspeed is 510 km/hr, so its velocity components are:
- Eastward component: \( 510 \times \cos(\theta) \)
- Northward component: \( 510 \times \sin(\theta) \)

Since the plane needs to end up going due east, the northward and southward components must cancel each other out:
\[
510 \times \sin(\theta) = 42.43
\]
Now, solve for \( \theta \):
\[
\sin(\theta) = \frac{42.43}{510} \approx 0.0832
\]
\[
\theta = \arcsin(0.0832) \approx 4.77^\circ
\]
So, the airplane should head \( \mathbf{4.77^\circ} \) **north of east**.

### Step 3: Calculate the Airplane's Speed Relative to the Ground
The ground speed is determined by the eastward components of both the airplane's velocity and the wind:
\[
\text{Ground speed} = 510 \times \cos(\theta) + 42.43
\]
Substituting \( \theta = 4.77^\circ \):
\[
\text{Ground speed} = 510 \times \cos(4.77^\circ) + 42.43 \approx 510 \times 0.9964 + 42.43 \approx 551 \text{ km/hr}
\]

### Final Answers:
1. **Direction:** The airplane should head \( \mathbf{4.77^\circ} \) **north of east**.
2. **Ground speed:** The airplane's speed relative to the ground is approximately \( \mathbf{551} \) **km/hr**.

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How would the airplane's heading change if the wind direction were northeast instead of southeast?
2. What would be the ground speed if the airplane were flying directly into the wind?
3. How does the wind's speed affect the airplane's optimal heading direction?
4. What would happen to the required heading angle if the wind speed increased?
5. How does the concept of vector addition apply to other real-world navigation problems?

**Tip:** When dealing with vector problems in navigation, breaking vectors into components along standard axes (east, north) simplifies calculations significantly.

Learn how to calculate the airplane's direction and speed relative to the ground when facing wind. This problem involves vector addition and trigonometric functions to determine the optimal heading direction and ground speed. Suitable for advanced high school students.

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