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### Part (a): Calculate the distance between Ships A and B
1. Convert the bearings from degrees to radians:
- Bearing of Ship A: 258°
[tex]\[ 258^\circ \times \left(\frac{\pi}{180}\right) = 4.50294947014537 \text{ radians} \][/tex]
- Bearing of Ship B: 312°
[tex]\[ 312^\circ \times \left(\frac{\pi}{180}\right) = 5.445427266222308 \text{ radians} \][/tex]
2. Calculate the coordinates of Ship A from point X:
- Distance to Ship A = 50 km
- Using trigonometric functions (cosine and sine) to find [tex]\(x_A\)[/tex] and [tex]\(y_A\)[/tex]:
[tex]\[ x_A = 50 \times \cos(4.50294947014537) = -10.40 \text{ km} \][/tex]
[tex]\[ y_A = 50 \times \sin(4.50294947014537) = -48.91 \text{ km} \][/tex]
3. Calculate the coordinates of Ship B from point X:
- Distance to Ship B = 44 km
- Using trigonometric functions (cosine and sine) to find [tex]\(x_B\)[/tex] and [tex]\(y_B\)[/tex]:
[tex]\[ x_B = 44 \times \cos(5.445427266222308) = 29.44 \text{ km} \][/tex]
[tex]\[ y_B = 44 \times \sin(5.445427266222308) = -32.70 \text{ km} \][/tex]
4. Find the distance between Ships A and B using the Pythagorean theorem:
- The differences in coordinates:
[tex]\[ \Delta x = x_B - x_A = 29.44 - (-10.40) = 39.84 \text{ km} \][/tex]
[tex]\[ \Delta y = y_B - y_A = -32.70 - (-48.91) = 16.21 \text{ km} \][/tex]
- Distance between the ships [tex]\(AB\)[/tex]:
[tex]\[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(29.44 + 10.40)^2 + (-32.70 + 48.91)^2} = 43.00866063844721 \text{ km} \][/tex]
### Part (b): Calculate the bearing of A from B
1. Find the difference in coordinates between Ship A and Ship B:
- [tex]\( \Delta x = x_A - x_B = -10.40 - 29.44 = -39.84 \text{ km} \)[/tex]
- [tex]\( \Delta y = y_A - y_B = -48.91 - (-32.70) = -16.21 \text{ km} \)[/tex]
2. Calculate the initial bearing in radians:
- Bearing [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{\Delta y}{\Delta x}\right) = \arctan\left(\frac{-16.21}{-39.84}\right) = 3.527793479688554 \text{ radians} \][/tex]
3. Convert the bearing from radians to degrees:
- Bearing in degrees:
[tex]\[ \text{Bearing} = 3.527793479688554 \times \left(\frac{180}{\pi}\right) = 202.1404175565383^\circ \][/tex]
Thus, the coordinates of Ships A and B are approximately [tex]\((-10.40, -48.91)\)[/tex] and [tex]\(29.44, -32.70)\)[/tex] respectively. The distance between the ships is approximately 43.01 km and the bearing of A from B is approximately 202.14°.
### Part (a): Calculate the distance between Ships A and B
1. Convert the bearings from degrees to radians:
- Bearing of Ship A: 258°
[tex]\[ 258^\circ \times \left(\frac{\pi}{180}\right) = 4.50294947014537 \text{ radians} \][/tex]
- Bearing of Ship B: 312°
[tex]\[ 312^\circ \times \left(\frac{\pi}{180}\right) = 5.445427266222308 \text{ radians} \][/tex]
2. Calculate the coordinates of Ship A from point X:
- Distance to Ship A = 50 km
- Using trigonometric functions (cosine and sine) to find [tex]\(x_A\)[/tex] and [tex]\(y_A\)[/tex]:
[tex]\[ x_A = 50 \times \cos(4.50294947014537) = -10.40 \text{ km} \][/tex]
[tex]\[ y_A = 50 \times \sin(4.50294947014537) = -48.91 \text{ km} \][/tex]
3. Calculate the coordinates of Ship B from point X:
- Distance to Ship B = 44 km
- Using trigonometric functions (cosine and sine) to find [tex]\(x_B\)[/tex] and [tex]\(y_B\)[/tex]:
[tex]\[ x_B = 44 \times \cos(5.445427266222308) = 29.44 \text{ km} \][/tex]
[tex]\[ y_B = 44 \times \sin(5.445427266222308) = -32.70 \text{ km} \][/tex]
4. Find the distance between Ships A and B using the Pythagorean theorem:
- The differences in coordinates:
[tex]\[ \Delta x = x_B - x_A = 29.44 - (-10.40) = 39.84 \text{ km} \][/tex]
[tex]\[ \Delta y = y_B - y_A = -32.70 - (-48.91) = 16.21 \text{ km} \][/tex]
- Distance between the ships [tex]\(AB\)[/tex]:
[tex]\[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(29.44 + 10.40)^2 + (-32.70 + 48.91)^2} = 43.00866063844721 \text{ km} \][/tex]
### Part (b): Calculate the bearing of A from B
1. Find the difference in coordinates between Ship A and Ship B:
- [tex]\( \Delta x = x_A - x_B = -10.40 - 29.44 = -39.84 \text{ km} \)[/tex]
- [tex]\( \Delta y = y_A - y_B = -48.91 - (-32.70) = -16.21 \text{ km} \)[/tex]
2. Calculate the initial bearing in radians:
- Bearing [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{\Delta y}{\Delta x}\right) = \arctan\left(\frac{-16.21}{-39.84}\right) = 3.527793479688554 \text{ radians} \][/tex]
3. Convert the bearing from radians to degrees:
- Bearing in degrees:
[tex]\[ \text{Bearing} = 3.527793479688554 \times \left(\frac{180}{\pi}\right) = 202.1404175565383^\circ \][/tex]
Thus, the coordinates of Ships A and B are approximately [tex]\((-10.40, -48.91)\)[/tex] and [tex]\(29.44, -32.70)\)[/tex] respectively. The distance between the ships is approximately 43.01 km and the bearing of A from B is approximately 202.14°.
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