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Sagot :
Certainly! To solve this problem of determining the distances travelled East and South by a ship navigating along specified bearings and distances, we can decompose each journey into its respective eastward and southward components. Let's go through each part step-by-step:
### Part a: Bearing [tex]\(132^\circ\)[/tex] for 100 km
1. Convert the bearing angle to a more familiar reference: The bearing [tex]\(132^\circ\)[/tex] is measured clockwise from the North. This corresponds to [tex]\(180^\circ - 132^\circ = 48^\circ\)[/tex] away from due South, or equivalently, [tex]\(132^\circ - 90^\circ = 42^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component can be found using the sine function: [tex]\(100 \sin(132^\circ)\)[/tex].
- The Southward component can be found using the cosine function: [tex]\(100 \cos(132^\circ)\)[/tex].
Given the previously derived values, we can state:
- Eastward distance: Approximately [tex]\(74.31\)[/tex] km East
- Southward distance: Approximately [tex]\(-66.91\)[/tex] km South
### Part b: Bearing [tex]\(151^\circ\)[/tex] for 50 km
1. Bearing conversion: The bearing [tex]\(151^\circ\)[/tex] is measured clockwise from the North. This corresponds to [tex]\(180^\circ - 151^\circ = 29^\circ\)[/tex] away from due South, or equivalently, [tex]\(151^\circ - 90^\circ = 61^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component: [tex]\(50 \sin(151^\circ)\)[/tex].
- The Southward component: [tex]\(50 \cos(151^\circ)\)[/tex].
Given the values:
- Eastward distance: Approximately [tex]\(24.24\)[/tex] km East
- Southward distance: Approximately [tex]\(-43.73\)[/tex] km South
### Part c: Bearing [tex]\(165^\circ\)[/tex] for 20 km
1. Bearing conversion: The bearing [tex]\(165^\circ\)[/tex] corresponds to [tex]\(180^\circ - 165^\circ = 15^\circ\)[/tex] away from due South, or equivalently, [tex]\(165^\circ - 90^\circ = 75^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component: [tex]\(20 \sin(165^\circ)\)[/tex].
- The Southward component: [tex]\(20 \cos(165^\circ)\)[/tex].
Given the values:
- Eastward distance: Approximately [tex]\(5.18\)[/tex] km East
- Southward distance: Approximately [tex]\(-19.32\)[/tex] km South
### Summary of Results
- For [tex]\(132^\circ\)[/tex] bearing and 100 km:
- East: [tex]\(74.31\)[/tex] km
- South: [tex]\(-66.91\)[/tex] km
- For [tex]\(151^\circ\)[/tex] bearing and 50 km:
- East: [tex]\(24.24\)[/tex] km
- South: [tex]\(-43.73\)[/tex] km
- For [tex]\(165^\circ\)[/tex] bearing and 20 km:
- East: [tex]\(5.18\)[/tex] km
- South: [tex]\(-19.32\)[/tex] km
These results provide the detailed distances the ship travels in the East and South directions for each segment of the journey following the given bearings and distances.
### Part a: Bearing [tex]\(132^\circ\)[/tex] for 100 km
1. Convert the bearing angle to a more familiar reference: The bearing [tex]\(132^\circ\)[/tex] is measured clockwise from the North. This corresponds to [tex]\(180^\circ - 132^\circ = 48^\circ\)[/tex] away from due South, or equivalently, [tex]\(132^\circ - 90^\circ = 42^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component can be found using the sine function: [tex]\(100 \sin(132^\circ)\)[/tex].
- The Southward component can be found using the cosine function: [tex]\(100 \cos(132^\circ)\)[/tex].
Given the previously derived values, we can state:
- Eastward distance: Approximately [tex]\(74.31\)[/tex] km East
- Southward distance: Approximately [tex]\(-66.91\)[/tex] km South
### Part b: Bearing [tex]\(151^\circ\)[/tex] for 50 km
1. Bearing conversion: The bearing [tex]\(151^\circ\)[/tex] is measured clockwise from the North. This corresponds to [tex]\(180^\circ - 151^\circ = 29^\circ\)[/tex] away from due South, or equivalently, [tex]\(151^\circ - 90^\circ = 61^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component: [tex]\(50 \sin(151^\circ)\)[/tex].
- The Southward component: [tex]\(50 \cos(151^\circ)\)[/tex].
Given the values:
- Eastward distance: Approximately [tex]\(24.24\)[/tex] km East
- Southward distance: Approximately [tex]\(-43.73\)[/tex] km South
### Part c: Bearing [tex]\(165^\circ\)[/tex] for 20 km
1. Bearing conversion: The bearing [tex]\(165^\circ\)[/tex] corresponds to [tex]\(180^\circ - 165^\circ = 15^\circ\)[/tex] away from due South, or equivalently, [tex]\(165^\circ - 90^\circ = 75^\circ\)[/tex] from the East towards the South.
2. Eastward and Southward components:
- The Eastward component: [tex]\(20 \sin(165^\circ)\)[/tex].
- The Southward component: [tex]\(20 \cos(165^\circ)\)[/tex].
Given the values:
- Eastward distance: Approximately [tex]\(5.18\)[/tex] km East
- Southward distance: Approximately [tex]\(-19.32\)[/tex] km South
### Summary of Results
- For [tex]\(132^\circ\)[/tex] bearing and 100 km:
- East: [tex]\(74.31\)[/tex] km
- South: [tex]\(-66.91\)[/tex] km
- For [tex]\(151^\circ\)[/tex] bearing and 50 km:
- East: [tex]\(24.24\)[/tex] km
- South: [tex]\(-43.73\)[/tex] km
- For [tex]\(165^\circ\)[/tex] bearing and 20 km:
- East: [tex]\(5.18\)[/tex] km
- South: [tex]\(-19.32\)[/tex] km
These results provide the detailed distances the ship travels in the East and South directions for each segment of the journey following the given bearings and distances.
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