The bearing of B from A is 210° and the bearing of C from B is 150° C calculate the distance between A and C to the nearest kilometers if AB=50 kilometers and BC=80 kilometers, also calculate the bearing of A and C to the nearest degree.
To find the distance between A and C, we can use the Law of Cosines. Let's label the angle between AB and BC as angle B.
Using the Law of Cosines, we have:
AC² = AB² + BC² - 2(AB)(BC)cos(B)
AC² = (50)² + (80)² - 2(50)(80)cos(150°)
AC² = 2500 + 6400 - 8000cos(150°)
AC² = 8900 - 8000(-0.866)
AC² = 8900 + 6928
AC² = 15828
Taking the square root of both sides, we get:
AC ≈ √15828 ≈ 125.89 kilometers
Therefore, the distance between A and C is approximately 125.89 kilometers.
To find the bearing of A and C, we can use the angles in the triangle.
The bearing of A from C is the sum of the bearing of C from B and the bearing of B from A.
Bearing of A from C = Bearing of C from B + Bearing of B from A
Bearing of A from C = 150° + 210°
Bearing of A from C = 360°
Since the bearing is measured clockwise from the North, we can subtract 360° to find the bearing to the nearest degree.
Bearing of A from C = 360° - 360°
Bearing of A from C = 0°
Therefore, the bearing of A from C is approximately 0°.