To determine the southern component of Jin's displacement, we need to break the displacement into its components based on the given angle and magnitude. Here's a step-by-step explanation:
1. Understand the problem:
- Jin walked 4 km at an angle of [tex]\( 30^\circ \)[/tex] south of east.
- We need to find the southern component of this displacement.
2. Visualize the scenario:
- Draw the direction of Jin's displacement.
- The displacement makes an angle of [tex]\( 30^\circ \)[/tex] with the east direction and points towards the south.
3. Component analysis:
- The displacement can be resolved into two perpendicular components: eastward (x-axis) and southward (y-axis).
4. Trigonometric relationship:
- The southern component (y-direction) of displacement can be calculated using the sine of the angle since sine relates the opposite side (south component) to the hypotenuse (total displacement).
5. Set up the equation:
- The magnitude of the displacement is 4 km.
- The angle given is [tex]\( 30^\circ \)[/tex].
6. Apply the sine function:
- [tex]\( \text{Southern component} = 4 \times \sin(30^\circ) \)[/tex]
7. Calculate the sine value:
- [tex]\( \sin(30^\circ) = 0.5 \)[/tex]
8. Compute the southern component:
- [tex]\( \text{Southern component} = 4 \times 0.5 = 2 \)[/tex] km
9. Round to the nearest tenth:
- The unrounded southern component is approximately [tex]\( 2.0 \)[/tex] km.
- After rounding, the southern component is [tex]\( 2.0 \)[/tex] km.
10. Select the correct answer:
- The southern component rounded to the nearest tenth is [tex]\( 2.0 \)[/tex] km.
- Among the given options, [tex]\( 2 \)[/tex] km is correct.
Therefore, the southern component of Jin's displacement, rounded to the nearest tenth, is:
[tex]\[ \boxed{2 \text{ km}} \][/tex]
