an airplane has an airspeed of 430 miles per hour at a bearing of 135 degrees. the wind velocity is 35 miles per hour in the direction N30degreesE. Find the resultant speed and direction of the plane.

Mathematicians should not write navigation problems:

an airplane has an airspeed of 430 miles per hour ***ON*** a ****HEADING**** of 135 degrees. the wind velocity is 35 miles per hour in the direction N30degreesE. Find the resultant speed and direction of the plane.
Usually we say where the wind is FROM but this seems to imply that the air is moving N 30 E . Again mathematicians do not get this stuff.

North speed = 430 cos 135 + 35 cos 30
= -273 mph or 273 south

East speed = 430 sin 135 + 35 sin 30
= 322 mph

total speed = sqrt(273^2+322^2) = 422 mph

angle of travel east of south = A
tan A = 322/273
A = 49.7
so actual direction over ground is
180 - 49.7 = 130.3 degrees East of North

2045.6

To find the resultant speed and direction of the plane, we need to use vector addition.

1. Convert the given airspeed and wind velocity into vector form:
- Airspeed: 430 mph at a bearing of 135 degrees
- Convert to vector form: (430 * cos(135°), 430 * sin(135°))
- Simplify: (-304.1, 304.1)
- Wind Velocity: 35 mph at a direction of N30°E
- Convert to vector form: (35 * cos(120°), 35 * sin(120°))
- Simplify: (-17.5, 30.3)

2. Add the airspeed vector and wind velocity vector to obtain the resultant vector:
Resultant Vector = Airspeed Vector + Wind Velocity Vector
Resultant Vector = (-304.1, 304.1) + (-17.5, 30.3)

Adding the corresponding components of the vectors:
X-component of Resultant Vector = -304.1 - 17.5 = -321.6
Y-component of Resultant Vector = 304.1 + 30.3 = 334.4

Therefore, the resultant vector is (-321.6, 334.4).

3. Calculate the magnitude (speed) and direction of the resultant vector:
Magnitude (speed) = sqrt((-321.6)^2 + (334.4)^2)
Magnitude (speed) ≈ 460.4 mph

Direction = arctan((Y-component / X-component)) + 180°
Direction = arctan(334.4 / -321.6) + 180°
Direction ≈ 137.5° + 180°

Therefore, the resultant speed of the plane is approximately 460.4 mph, and the direction is approximately 317.5°.

To find the resultant speed and direction of the plane, we need to apply vector addition.

Step 1: Resolve the wind velocity into its respective components.
The wind velocity is given as 35 miles per hour in the direction N30°E. We can use trigonometry to resolve it into its northward and eastward components.

N30°E forms a right triangle, with the northward direction being the adjacent side and the eastward direction being the opposite side.
By using sine and cosine functions, we can find the respective components:

Northward component = 35 * sin(30°)
Eastward component = 35 * cos(30°)

Step 2: Calculate the components of the resultant velocity.
The airplane's airspeed is given as 430 miles per hour at a bearing of 135°, which means it has components in the northward and eastward directions. We can use trigonometry to find these components.

Northward component = 430 * sin(135°)
Eastward component = 430 * cos(135°)

Step 3: Add the respective components to find the resultant components.
To find the resultant velocity in each direction, we need to add the corresponding components of the airplane's airspeed and wind velocity.

Resultant northward velocity = Airplane's northward component + Wind's northward component
Resultant eastward velocity = Airplane's eastward component + Wind's eastward component

Step 4: Calculate the magnitude and direction of the resultant velocity.
To find the magnitude of the resultant velocity, we can use the Pythagorean theorem with the resultant northward and eastward velocities.

Magnitude of resultant velocity = sqrt((Resultant northward velocity)^2 + (Resultant eastward velocity)^2)

The direction of the resultant velocity can be found using the inverse tangent function:

Direction of resultant velocity = tan^(-1)(Resultant northward velocity / Resultant eastward velocity)

By plugging in the respective values and performing the calculations, we can find the resultant speed and direction of the plane.