Sure, let's solve the problem step-by-step.
### Part (a): How far south has it travelled?
1. Identify the given information:
- The ship travels a distance of [tex]\( 200 \)[/tex] km.
- The bearing is [tex]\( 243.7^\circ \)[/tex].
2. Understanding the Bearing:
- A bearing of [tex]\( 243.7^\circ \)[/tex] is measured clockwise from the north direction.
- To calculate the southward (S) distance, we need to resolve the distance into its southward component.
3. Using Trigonometry:
- We can use sine function to determine the southward component.
- Mathematically, the southward distance [tex]\( S \)[/tex] can be found using:
[tex]\[
S = 200 \times \sin(243.7^\circ)
\][/tex]
4. Calculation:
According to our given data:
[tex]\[
S \approx -179.297 \text{ km}
\][/tex]
So, the ship has travelled approximately [tex]\( -179.297 \)[/tex] km south, where the negative sign indicates a direction.
### Part (b): How far west has it travelled?
1. Identify the given information:
- The same distance and bearing as in part (a).
2. Using Trigonometry:
- We can use the cosine function to determine the westward component.
- Mathematically, the westward distance [tex]\( W \)[/tex] can be found using:
[tex]\[
W = 200 \times \cos(243.7^\circ)
\][/tex]
3. Calculation:
According to our given data:
[tex]\[
W \approx -88.614 \text{ km}
\][/tex]
So, the ship has travelled approximately [tex]\( -88.614 \)[/tex] km west, where the negative sign indicates a direction.
### Summary:
(a) The ship has travelled approximately [tex]\( 179.297 \)[/tex] km south.
(b) The ship has travelled approximately [tex]\( 88.614 \)[/tex] km west.
Note that in the context of travel distances, we often consider absolute values. So, the southward and westward distances are:
(a) 179.297 km south.
(b) 88.614 km west.
