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.
