A ship sails 133 miles north, then turns and sails at an angle of [tex]$20^{\circ}$[/tex] east of north for 229 miles. What is the direction of the ship's resultant vector?

[tex]
\begin{array}{c}
|\overrightarrow{ R }| = 356.89 \text { miles } \\
\theta = [?]^{\circ}
\end{array}
[/tex]

Round your answer to the nearest hundredth.



Answer :

To find the direction of the ship's resultant vector when it sails 133 miles north and then turns and sails at an angle of [tex]$20^{\circ}$[/tex] east of north for 229 miles, we need to break down the motion into its components and then use trigonometry to find the resultant direction.

### Step-by-Step Solution:

#### 1. Break Down the Second Leg into Components:

The ship sails 229 miles at an angle of [tex]$20^{\circ}$[/tex] east of north. This direction forms a triangle where:

- x-component (East/West): [tex]\( 229 \sin(20^{\circ}) \)[/tex]
- y-component (North/South): [tex]\( 229 \cos(20^{\circ}) \)[/tex]

Given the calculations, these components are:
- x-component: 78.32 miles (east)
- y-component: 215.19 miles (north)

#### 2. Sum up the Northward Components:

The ship initially sails 133 miles north. Adding this to the y-component of the second leg:

Total northward distance: [tex]\(133 + 215.19 = 348.19\)[/tex] miles

#### 3. Calculate the Direction:

To find the angle [tex]$\theta$[/tex] north of east, we use the arctangent of the ratio of the x-component to the total y-component. The formula for [tex]$\theta$[/tex] is:

[tex]\[ \theta = \arctan\left(\frac{\text{x-component}}{\text{total y-component}}\right) \][/tex]

Substituting the given values:

[tex]\[ \theta = \arctan\left(\frac{78.32}{348.19}\right) \][/tex]

#### 4. Convert the Angle to Degrees:

Using a calculator or a trigonometric table, we find:

[tex]\[ \theta \approx 12.68^\circ \][/tex]

Thus, the direction of the ship's resultant vector is approximately [tex]\( \boxed{12.68^\circ} \)[/tex] east of north.