An airplane is flying on a compass heading of 160 degrees at 425 mph. A wind is blowing with the bearing 200 degrees at 75 mph. Find the actual speed and direction of the airplane.

Vr = 425mph[160o] + 75mph[200o].

X = 425*cos160+75*cos200 = -470 mph.
Y = 425*sin160+75*sin200 = 120 mph.

Tan A = Y/X = 120/-470 = -0.25532.
A = -14.3o. = 14.3o N. of W. = The direction.

Vr = -470/-Cos14.3 = 485 mph. = Resultant speed.

To find the actual speed and direction of the airplane, we need to calculate the components of the velocity vector for both the airplane and the wind, and then add them together.

Let's start by breaking down the airplane's velocity into its components. We'll use the following convention:

x-axis: East/West direction (positive to the East)
y-axis: North/South direction (positive to the North)

Given:
Airplane heading = 160 degrees
Airplane speed = 425 mph

To find the components of the airplane's velocity, we'll use trigonometry. The cosine of the heading angle gives us the component in the x-direction (East/West), and the sine of the heading angle gives us the component in the y-direction (North/South).

Component in x-direction (Vx):
Vx = airplane speed * cos(heading angle)
= 425 * cos(160)

Component in y-direction (Vy):
Vy = airplane speed * sin(heading angle)
= 425 * sin(160)

Next, let's break down the wind velocity in a similar way.

Given:
Wind bearing = 200 degrees
Wind speed = 75 mph

To find the components of the wind velocity, we'll again use trigonometry. Since the bearing angle is measured clockwise from the North, we need to subtract it from 90 degrees to find the angle with respect to the x-axis.

Component in x-direction (Wx):
Wx = wind speed * cos(90 - bearing angle)
= 75 * cos(90 - 200)

Component in y-direction (Wy):
Wy = wind speed * sin(90 - bearing angle)
= 75 * sin(90 - 200)

Now, we can find the total velocity of the airplane by adding the components of the airplane's velocity and the wind's velocity.

Total x-component (Ax):
Ax = Vx + Wx

Total y-component (Ay):
Ay = Vy + Wy

To find the magnitude of the total velocity (V) and its direction (θ), we can use Pythagoras' theorem and inverse trigonometric functions.

Total velocity (V):
V = √(Ax² + Ay²)

Direction (θ):
θ = arctan(Ay / Ax)

Calculating the values:

Vx = 425 * cos(160) = -338.41 mph (negative because it's to the West)
Vy = 425 * sin(160) = 198.80 mph (positive to the North)

Wx = 75 * cos(90 - 200) = -72.51 mph (negative because it's to the West)
Wy = 75 * sin(90 - 200) = -2.20 mph (negative because it's to the South)

Ax = Vx + Wx = -338.41 - 72.51 = -410.92 mph (negative because it's to the West)
Ay = Vy + Wy = 198.80 - 2.20 = 196.60 mph (positive to the North)

V = √(Ax² + Ay²) = √((-410.92)² + 196.60²) = 456.06 mph
θ = arctan(Ay / Ax) = arctan(196.60 / -410.92) = -25.34 degrees

Therefore, the actual speed of the airplane is approximately 456.06 mph and its actual direction is approximately 25.34 degrees to the West of North.

To find the actual speed and direction of the airplane, we need to consider the effect of the wind on the plane's motion. We can break down the wind's velocity into its north-south and east-west components using trigonometry.

First, let's consider the wind's north-south component. The wind is blowing with a bearing of 200 degrees, which means it is coming from the south towards the north. We can find the north-south component of the wind's velocity by multiplying the wind speed (75 mph) by the sine of the angle between the northern direction and the wind's bearing.

So, the north-south component of the wind's velocity is:
North-South Component = 75 mph * sin(180 degrees - 200 degrees)

Next, let's consider the wind's east-west component. The wind is blowing with a bearing of 200 degrees, so it is coming from the west towards the east. We can find the east-west component of the wind's velocity by multiplying the wind speed (75 mph) by the cosine of the angle between the western direction and the wind's bearing.

So, the east-west component of the wind's velocity is:
East-West Component = 75 mph * cos(180 degrees - 200 degrees)

Now, let's find the net north-south and east-west velocities by subtracting the wind's components from the plane's velocity. Since the plane is flying on a compass heading of 160 degrees at 425 mph, its north-south component of velocity is given by multiplying the plane's speed by the cosine of the difference between the plane's heading and the northern direction (90 degrees).

So, the north-south component of the plane's velocity is:
North-South Component = 425 mph * cos(90 degrees - 160 degrees)

Similarly, the east-west component of the plane's velocity can be found by multiplying its speed by the sine of the difference between the plane's heading and the northern direction.

So, the east-west component of the plane's velocity is:
East-West Component = 425 mph * sin(90 degrees - 160 degrees)

Finally, we can calculate the actual speed and direction of the plane using the net north-south and east-west components of the velocity. The actual speed of the plane is the magnitude of the resultant velocity, which can be found using the Pythagorean theorem:
Actual Speed = sqrt((North-South Component + North-South Component of Wind)^2 + (East-West Component + East-West Component of Wind)^2)

The actual direction of the plane is the inverse tangent of the net north-south and east-west components of the velocity:
Actual Direction = arctan((East-West Component + East-West Component of Wind) / (North-South Component + North-South Component of Wind))

By plugging in the values and calculating these expressions, you can find the actual speed and direction of the airplane.