An aircraft left a Port X on s bearing of 105 degree to Y a distance of 40km. At Y it changed direction and flew to another Port Z, 50km away, until it was due south of it's starting point. Calculate correct to 2 significant figures; a) distance /XZ/
b) the bearing of X from Y.
To solve this problem, we can use trigonometry and vector addition. Let's break it down step by step:
a) Distance XZ:
To find the distance XZ, we can use the concept of vector addition. We know the distance XY is 40km, and the distance YZ is 50km. Now we need to find the distance XZ.
First, let's find the north-south component of the XZ vector. Since the aircraft flies due south of the starting point, this component is equal to the distance XY. So, the north-south component of XZ is 40km.
Next, we need to find the east-west component of the XZ vector. To do this, we can use trigonometry. Since the aircraft changed direction by 180 - 105 = 75 degrees at Y, we can consider the triangle formed by XY and XZ as a right triangle. The east-west component is the adjacent side to the angle of 75 degrees, and the hypotenuse is the distance XZ.
Using trigonometry, we can use the cosine function to find the east-west component:
cos(angle) = adjacent / hypotenuse
cos(75) = east-west component / XZ
east-west component = cos(75) * XZ
Now, we can set up an equation using the given distances:
east-west component = cos(75) * XZ = XY = 40km
Solving for XZ:
cos(75) * XZ = 40km
XZ = 40km / cos(75)
Calculating XZ:
XZ ≈ 40km / 0.2588 ≈ 154.60km
Therefore, the distance XZ is approximately 154.60km.
b) Bearing of X from Y:
To find the bearing of X from Y, we need to consider the triangle formed by XY and XZ. The bearing is the angle measured clockwise from the north direction.
Since the aircraft initially flew from X to Y on a bearing of 105 degrees, we need to find the angle between XY and the north direction.
Let's call this angle α. We can use trigonometry to find α:
sin(α) = opposite / hypotenuse
sin(α) = east-west component of XZ / XZ
sin(α) = cos(75) * XZ / XZ
sin(α) = cos(75)
Solving for α:
α ≈ arcsin(cos(75))
Calculating α:
α ≈ arcsin(0.2588)
α ≈ 15.4649 degrees
Since the bearing is measured clockwise from the north direction, the bearing of X from Y is 360 - α degrees.
Calculating the bearing:
Bearing of X from Y ≈ 360 - 15.4649 ≈ 344.53 degrees
Therefore, the bearing of X from Y is approximately 344.53 degrees.