Join IDNLearn.com today and start getting the answers you've been searching for. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.
Sagot :
Sure, let's solve this step by step.
### 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°.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
