a plane is heading 60 degrees east of south at 180 km/hr. a 50 km/hr wind blows in the direction of 25 degrees west of south. find the resultant velocity and bearing.

Make a reasonable sketch.

Using vectors and using R to be the resultant vector:

R = (180cos300,180sin300) + (50cos245,50sin245)
= (90,-90√3) + (-21.1309.. , -45.31538..)
= (68.869.., -201.1996...)

Magnitude = √(68.869^2 + (-201.1996..)^2)
= appr 212.66 km

direction:
tanØ = -201.1996/68.869
= ..
Ø = 360-71.104
= appr 288.9°

check my arithmetic

To find the resultant velocity and bearing, we need to find the vectors of the plane's velocity and the wind's velocity and then add them together.

Step 1: Break down the plane's velocity into its components:

The plane is heading 60 degrees east of south, which means it is moving in the direction of 240 degrees (180 + 60) from the positive x-axis.

The magnitude of the plane's velocity is 180 km/hr. The x-component can be found using the cosine function:

x-component = 180 km/hr * cos(240 degrees) = -90 km/hr

The y-component can be found using the sine function:

y-component = 180 km/hr * sin(240 degrees) = -155.88 km/hr

So the plane's velocity vector is (-90 km/hr, -155.88 km/hr).

Step 2: Break down the wind's velocity into its components:

The wind is blowing in the direction of 25 degrees west of south, which means it is moving in the direction of 205 degrees (180 - 25) from the positive x-axis.

The magnitude of the wind's velocity is 50 km/hr. The x-component can be found using the cosine function:

x-component = 50 km/hr * cos(205 degrees) = -34.73 km/hr

The y-component can be found using the sine function:

y-component = 50 km/hr * sin(205 degrees) = -21.43 km/hr

So the wind's velocity vector is (-34.73 km/hr, -21.43 km/hr).

Step 3: Add the two velocity vectors together:

To find the resultant velocity, add the x-components and the y-components separately:

Resultant x-component = -90 km/hr + (-34.73 km/hr) = -124.73 km/hr

Resultant y-component = -155.88 km/hr + (-21.43 km/hr) = -177.31 km/hr

So the resultant velocity vector is (-124.73 km/hr, -177.31 km/hr).

Step 4: Find the magnitude and direction of the resultant velocity:

The magnitude of the resultant velocity can be found using the Pythagorean theorem:

Resultant magnitude = sqrt((-124.73 km/hr)^2 + (-177.31 km/hr)^2) = 218.47 km/hr

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

Resultant direction = atan2(-177.31 km/hr, -124.73 km/hr) + 180 degrees = 238.90 degrees

Note: atan2 is used to account for the correct quadrant of the resultant vector.

So the resultant velocity is 218.47 km/hr in the direction of 238.90 degrees from the positive x-axis.

To find the resultant velocity and bearing, we need to consider the vector addition of the velocity of the plane and the velocity of the wind.

Let's break down the velocity of the plane and the wind into their respective vector components.

Velocity of the plane:
- Magnitude: 180 km/hr
- Direction: 60 degrees east of south

Using trigonometry, we can find the vector components of the plane's velocity by resolving the magnitude and direction into their horizontal and vertical components.

Horizontal component:
180 km/hr * cos(60°) = 90 km/hr (east)

Vertical component:
180 km/hr * sin(60°) = 155.88 km/hr (south)

So, the vector components of the plane's velocity are 90 km/hr east and 155.88 km/hr south.

Velocity of the wind:
- Magnitude: 50 km/hr
- Direction: 25 degrees west of south

Again, let's find the vector components of the wind's velocity.

Horizontal component:
50 km/hr * cos(25°) = 45.37 km/hr (west)

Vertical component:
50 km/hr * sin(25°) = 21.27 km/hr (south)

Thus, the vector components of the wind's velocity are 45.37 km/hr west and 21.27 km/hr south.

Next, we need to calculate the resultant of the two vectors by adding their horizontal and vertical components separately.

Horizontal component of resultant velocity = (horizontal component of plane's velocity) + (horizontal component of wind's velocity)
= 90 km/hr (east) + 45.37 km/hr (west)
= 44.63 km/hr (east)

Vertical component of resultant velocity = (vertical component of plane's velocity) + (vertical component of wind's velocity)
= 155.88 km/hr (south) + 21.27 km/hr (south)
= 177.15 km/hr (south)

The resultant velocity of the plane and wind is 44.63 km/hr east and 177.15 km/hr south.

Finally, we can find the magnitude and direction (bearing) of the resultant velocity using trigonometry.

Magnitude: √((horizontal component)^2 + (vertical component)^2)
= √((44.63 km/hr)^2 + (177.15 km/hr)^2)
= √(1993.73 km^2/hr^2 + 31384.82 km^2/hr^2)
= √33378.55 km^2/hr^2
≈ 182.70 km/hr

Bearing: arctan((vertical component) / (horizontal component))
= arctan(177.15 km/hr / 44.63 km/hr)
= arctan(3.97)

Converting arctan to degrees: Bearing ≈ 78.32°

Therefore, the resultant velocity is approximately 182.70 km/hr at a bearing of 78.32°.