Question

An airplane is traveling due east with a velocity of 563 miles per hour. The wind blows at 76 miles per hour

at an angle of S43° . What is the resultant speed and direction of the plane?

Answer 1

The resultant speed of the plane is approximately 567.8 miles per hour, and the direction is approximately 5.7° south of east.

Step-by-step explanation:

In order to determine the resultant speed and direction of the plane, we need to break down the velocities into their components. The eastward velocity of the plane is 563 miles per hour, while the wind velocity has a magnitude of 76 miles per hour and is at an angle of S43° (43 degrees south of east).

To find the resultant speed, we can use the Pythagorean theorem. The eastward component of the wind's velocity can be found by multiplying the wind velocity by the cosine of the angle (43 degrees south of east) - this component is subtracted from the velocity of the plane. The northward component of the wind's velocity can be found by multiplying the wind velocity by the sine of the angle (43 degrees south of east).

Using these components, we can find the resultant speed (magnitude) and direction (angle) of the plane by using the Pythagorean theorem and the inverse tangent function, respectively.

Let's calculate the values:

1. The eastward component of the wind's velocity = 76 miles per hour * cos(43°) ≈ 58.15 miles per hour
2. The northward component of the wind's velocity = 76 miles per hour * sin(43°) ≈ 51.73 miles per hour
3. The resultant speed (magnitude) of the plane = sqrt((563 - 58.15)^2 + (0 - 51.73)^2) ≈ 567.8 miles per hour
4. The direction (angle) of the plane = atan((0 - 51.73)/(563 - 58.15)) ≈ -5.7° south of east

Therefore, the resultant speed of the plane is approximately 567.8 miles per hour, and the direction is approximately 5.7° south of east.