The bearing of point Y from a point X is 048°. If Y is 115km away from point X and the bearing of point Z from Y is 120° at a distance of 75km. Find the distance between X and Z, using a scale of 1cm to represent 20km
Did you make your sketch?
It should show an angle of 108° between XY and YZ
by the cosine law:
XZ^2 = 115^2 + 75^2 - 2(115)(75)cos 108°
XZ = .....
or
vector XY = 115(cos42, sin42) = (85.46165... , 76.95...)
vector YZ = 75(cos-30 , sin -30) = (64.9519.. , -37.5)
XZ = (150.4133.. , 39.45..)
|XZ| = √(150.4133..^2 + 39.45..^2) = .....
You must get the same answer, I did.
D = 115km[48o] + 75km[120o],
X = 115*sin48 + 75*sin120 = 150.4 km,
Y = 115*Cos48 + 75*Cos120 = 39.45 km,
D = 150.4 + 39.45i = 155.5km[75.3o] = Distance from X to Z.
Tan A = X/Y For bearing notation.
A = 75.3o.
To find the distance between point X and point Z, we need to determine the bearing from point X to point Z as well as the actual distance.
First, let's calculate the bearing from point X to point Z. We know the bearing from point Y to point Z is 120°. Since the bearing from point X to point Y is 048°, we can add the two bearings to find the bearing from X to Z.
The sum of the bearings is 048° + 120° = 168°.
Next, we need to find the actual distance between point X and point Z. We are given that the distance between point X and point Y is 115km, and we are given a scale of 1cm to represent 20km.
To convert the distance between X and Y to cm, we divide 115km by 20km/cm:
115km / 20km/cm = 5.75 cm
Now, using the scale, we can determine the actual distance between X and Z. We are given that 1cm on the map represents 20km, so to find the actual distance in km, we multiply the distance on the map (in cm) by the scale:
Distance between X and Z = 5.75 cm * 20km/cm = 115 km
So, the distance between point X and point Z is 115 km.