A ship leaves the island of Guam and sails a distance 265 km at an angle 41.0 north of west.

***In which direction must it now head so that its resultant displacement will be 130 km directly east of Guam? (Express your answer as an angle measured south of east)
***How far must it sail so that its resultant displacement will be 130 km directly east of Guam??

distance sailed+ new distance= 130E

265cos41 W + 265sin41 N + XX E + YY S =-130W

so on the left combine the Directons (remember W=-E), and S=-N
solve for XX and YY

To determine the direction the ship must now head, we need to find the angle south of east.

Step 1: Draw a diagram to visualize the problem.
Let the initial displacement be represented by vector A, and the final displacement be represented by vector B.

Step 2: Break down the initial displacement into its x and y components using trigonometry.
Given that the ship sails at an angle 41.0° north of west, we can find the x and y components as follows:
Ax = 265 km * cos(41.0°)
Ay = 265 km * sin(41.0°)

Step 3: Calculate the x and y components of the desired resultant displacement.
Since the desired resultant displacement is 130 km directly east of Guam, the x component should be 130 km, and the y component should be 0 km.

Step 4: Subtract the initial displacement components from the desired resultant displacement components to get the change in x and y.
Δx = 130 km - Ax
Δy = 0 km - Ay

Step 5: Calculate the angle south of east using the tangent function.
The angle can be found using the formula:
θ = arctan(Δy / Δx)

Substituting in the known values:
θ = arctan((0 km - Ay) / (130 km - Ax))

Now we can calculate the value of θ:

θ = arctan((- Ay) / (130 km - Ax))

To determine how far the ship must sail, we need to find the magnitude of the change in displacement.

Step 1: Use the Pythagorean theorem to find the magnitude of the change in displacement.
The magnitude can be found using the formula:
Magnitude of Δd = sqrt(Δx^2 + Δy^2)

Substituting in the known values:
Magnitude of Δd = sqrt(Δx^2 + Δy^2)

Now we can calculate the value of the magnitude.

To solve this problem, we can break it down into two parts: determining the direction and determining the distance.

1. Direction:
The ship initially sails at an angle of 41.0° north of west. If we want the resultant displacement to be 130 km directly east of Guam, the ship needs to head in a direction that will cancel out the initial westward component.

To find this direction, we can subtract the initial angle from 180° (since east is 180° from west).

180° - 41.0° = 139.0°

So, the ship must head in a direction 139.0° south of east to reach a resultant displacement directly east of Guam.

2. Distance:
To determine the distance the ship must sail, we can use the concept of vector addition. The initial displacement of 265 km at an angle of 41.0° north of west can be represented as a vector (negative since it is directed westward) as follows:

V1 = -265 km at 41.0°

The desired resultant displacement is 130 km directly east of Guam, which can be represented as:

V2 = 130 km at 0°

The resultant displacement is given by Vr = V1 + V2. We need to find the magnitude of Vr, which will give us the distance the ship must sail to achieve the desired displacement.

|Vr| = √(Vr_x^2 + Vr_y^2)

Since V1 and V2 are perpendicular to each other, the vector addition equation simplifies to:

|Vr| = |V1| + |V2|

Substituting the values, we get:

|Vr| = 265 km + 130 km
= 395 km

Therefore, the ship must sail a distance of 395 km to achieve a resultant displacement of 130 km directly east of Guam.

So, to summarize:
- The ship must head in a direction 139.0° south of east.
- The ship must sail a distance of 395 km.