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.