P, Q and R are points in the same horizontal plan. The bearing Q from P is 160 degrees and bearing of R from 050 degrees. If /PQ/= 6m and /QR/= 4m, find the /PR/ and bearing of R from P, correct to the nearest degree
Draw the diagram.
Use the law of cosines to get PR
Now you have a triangle RPQ
Then you can use either the law of cosines or sines to get the angle at P. The bearing of R from P will be 160+that angle
PR = PQ + QR.
PR = 6m[160o] + 4m[50o],
X = 6*sin160 + 4*sin50 = 5.12 m.
Y = 6*Cos160 + 4*Cos50 = -3.07 m.
PR = 5.12 - 3.07i = 5.97m[-59o] = 5.97m[121o] CW.
To find the length of PR and bearing of R from P, we can use the Law of Cosines and basic trigonometry.
First, let's draw a diagram to visualize the given information:
```
P_________Q
/
/
/
/
R
```
Given information:
- PQ = 6m
- QR = 4m
- The bearing of Q from P is 160 degrees (measured clockwise from the reference direction, e.g., North).
- The bearing of R from the reference direction is 050 degrees (measured clockwise from the reference direction).
Now, let's start by finding the length of PR.
Using the Law of Cosines, we have:
PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(angle PQR)
To find the angle PQR, we can subtract the bearing of Q from P from 180 degrees (bearing angle subtends to 180 degrees):
angle PQR = 180 degrees - 160 degrees = 20 degrees
Substituting the values, we get:
PR^2 = 6^2 + 4^2 - 2 * 6 * 4 * cos(20 degrees)
PR^2 = 36 + 16 - 48 * cos(20 degrees)
PR^2 ≈ 36 + 16 - 48 * 0.9397 (cos(20 degrees) ≈ 0.9397)
PR^2 ≈ 52 - 45.1184
PR^2 ≈ 6.8816
PR ≈ √6.8816
PR ≈ 2.623 meters
Therefore, the length of PR is approximately 2.623 meters.
Next, let's find the bearing of R from P.
Since we know the bearing of R from the reference direction is 050 degrees, and the bearing of Q from P is 160 degrees, we can find the bearing of R from P by subtracting the bearing of Q from P from 050 degrees:
Bearing of R from P = 050 degrees - 160 degrees = -110 degrees
To convert -110 degrees to a positive bearing (measured clockwise from the reference direction), we can add 360 degrees:
Bearing of R from P = -110 degrees + 360 degrees = 250 degrees
Therefore, the bearing of R from P is approximately 250 degrees.
In conclusion, the length of PR is approximately 2.623 meters, and the bearing of R from P is approximately 250 degrees.