A sailing ship left harbour A and travelled 20km on a bearing of 125 degrees. It then changed course and sailed on a bearing of 035 degrees on a distance of 10km until it reached B. Calculate the distance between A and B. The bearing of A from B and the bearing of B from A.
To calculate the distance between points A and B, we can use the cosine rule. Let's label the points as follows: A is the starting point, B is the ending point, and C is the vertex of the triangle formed by the two sides and the line connecting A and B.
First, let's calculate the length of side AC using the given information:
Side AC = √(20^2 + 10^2 - 2 * 20 * 10 * cos(180° - (125° + 35°)))
= √(400 + 100 - 400 * cos(180° - 160°))
= √(400 + 100 - 400 * cos(20°))
≈ √(500 - 400 * 0.9397)
≈ √(500 - 375.88)
≈ √(124.12)
≈ 11.14 km
Now, let's calculate the bearing of A from B:
Angle A = 180° - (125° + 180° - 35°)
= 30°
Finally, let's calculate the bearing of B from A:
Angle B = 360° - Angle A
= 360° - 30°
= 330°
Therefore, the distance between A and B is approximately 11.14 km. The bearing of A from B is 30°, and the bearing of B from A is 330°.
To find the distance between points A and B, you can use the cosine rule:
c² = a² + b² - 2ab * cos(C)
where a, b, and c are the sides of a triangle, and C is the angle opposite side c.
In this case, we can consider points A and B as the endpoints of the triangle, and the distance between them as the side c that we need to find. The bearings of A from B and B from A can be calculated using trigonometry.
1. Calculate the distance between A and B:
Using the cosine rule, let's calculate the distance "c" between points A and B:
c² = 20² + 10² - 2 * 20 * 10 * cos(180 - (125 - 35))
Simplifying:
c² = 20² + 10² - 2 * 20 * 10 * cos(180 - 90)
c² = 400 + 100 - 400 * 0
c² = 500
c = √500
c ≈ 22.36 km
Therefore, the distance between A and B is approximately 22.36 km.
2. Calculate the bearing of A from B:
The bearing of A from B is the same as the initial bearing of the ship when it left harbour A, which is given as 125 degrees.
Therefore, the bearing of A from B is 125 degrees.
3. Calculate the bearing of B from A:
The bearing of B from A can be calculated as the compass bearing needed to travel from A to B.
Since A is to the south of B, we can subtract the angle between points A and B (35 degrees) from 180 degrees to get the bearing of B from A:
Bearing of B from A = 180 - 35 = 145 degrees
Therefore, the bearing of B from A is 145 degrees.
In summary:
- The distance between A and B is approximately 22.36 km.
- The bearing of A from B is 125 degrees.
- The bearing of B from A is 145 degrees.