An airplane travels with a speed of 145 mph in calm air. the wind is blowing with a velocity of 23 mph from N72degreesE. If the plane is headed in a direction N63W, find the magnitude of the speed and the direction of the airplane with reference to the ground. In what direction must the airplane be headed in order to fly in the direction N63W, and what would be its actual speed in the air?

convert the two vectors p and w (fir plane and wind) to rectangular coordinates, and add them up.

p+w is the resultant velocity on the ground.

A vector w in the direction N63W, with magnitude w is <-0.454w,0.891w>.

To find the magnitude of the speed and the direction of the airplane with reference to the ground, we can use vector addition.

1. Start by decomposing the wind velocity into its north and west components. The north component can be found by multiplying the wind velocity (23 mph) by the cosine of the angle N72°E (since the x-axis is pointing east). The west component can be found by multiplying the wind velocity by the sine of the same angle.

North component of wind velocity = 23 mph * cos 72° = 8.535 mph
West component of wind velocity = 23 mph * sin 72° = 22.809 mph

2. Since the airplane is headed in a direction N63W, we can calculate the components of its velocity with respect to the ground. The north component can be found by multiplying the airplane's speed (145 mph) by the cosine of the angle N63°W. The west component can be found by multiplying the airplane's speed by the sine of the same angle.

North component of airplane's velocity = 145 mph * cos 63° = 68.269 mph
West component of airplane's velocity = 145 mph * sin 63° = 131.716 mph

3. Now, we can find the resultant of the wind and airplane velocities by adding their respective north and west components.

Resultant north component = north component of wind velocity + north component of airplane's velocity = 8.535 mph + 68.269 mph = 76.804 mph
Resultant west component = west component of wind velocity + west component of airplane's velocity = 22.809 mph + 131.716 mph = 154.525 mph

4. Finally, we can calculate the magnitude of the speed and the direction of the airplane with reference to the ground using the Pythagorean theorem and trigonometry.

Magnitude of speed = √((Resultant north component)^2 + (Resultant west component)^2) = √((76.804 mph)^2 + (154.525 mph)^2) = 172.264 mph

Direction of the airplane with reference to the ground = arctan(Resultant west component / Resultant north component) = arctan(154.525 mph / 76.804 mph) ≈ 63.16°

Therefore, the magnitude of the speed of the airplane with reference to the ground is approximately 172.264 mph, and the direction of the airplane with reference to the ground is approximately N63°E.

To fly in the direction N63W, the airplane must be headed in the opposite direction, which is S63°E.

The actual speed of the airplane in the air can be found using the Pythagorean theorem:

Actual speed in the air = √((Magnitude of speed)^2 - (wind velocity)^2) = √((172.264 mph)^2 - (23 mph)^2) ≈ 170.725 mph