Two ship P and Q left a port R at the same time.Q sailed on a bearing of 150 while P sailed on the north side of Q.After a distance of 8km and 10km by P and Q respectively,their dirtance part was 12km.find the bearing of P from R.
067degree
To find the bearing of ship P from port R, we can use trigonometry and the concept of bearing angles.
First, let's visualize the situation:
- Ship Q sailed on a bearing of 150 degrees.
- Ship P sailed on the north side of Q.
- After some distance, the distance between P and Q is 12km.
- Ship P traveled 8km, while Ship Q traveled 10km.
Let's break down the problem into two triangles: triangle PQR and triangle PQS, where S is the point where P and Q meet.
Triangle PQR:
- Q is at a distance of 10km from R.
- The angle QRP is 180 - 150 = 30 degrees (since the sum of the angles in a triangle is 180 degrees).
Triangle PQS:
- P is at a distance of 8km from R.
- The angle PQS is the bearing angle of P from Q.
Now, we can use the Law of Cosines to find the length of side QR (which equals 10km) in triangle PQR:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(QRP)
Substituting the given values:
10^2 = 8^2 + PR^2 - 2 * 8 * PR * cos(30)
Simplifying the equation:
100 = 64 + PR^2 - 16 * PR * (sqrt(3) / 2)
100 = 64 + PR^2 - 8 * PR * sqrt(3)
Rearranging the equation:
PR^2 - 8 * PR * sqrt(3) + 36 = 0
Now we can solve this quadratic equation:
Using the quadratic formula:
PR = ( -(-8 * sqrt(3)) ± sqrt( (-8 * sqrt(3))^2 - 4 * 1 * 36 ) ) / (2 * 1)
PR = ( 8 * sqrt(3) ± sqrt(192 - 144) ) / 2
PR = ( 8 * sqrt(3) ± sqrt(48) ) / 2
PR = 4 * sqrt(3) ± 4 * sqrt(3)
Since PR cannot be negative, we take the positive root:
PR = 4 * sqrt(3) + 4 * sqrt(3) = 8 * sqrt(3)
Now, to find the bearing of P from R, we need to find the angle RPQ.
Using the Law of Cosines in triangle PQR:
PQ^2 = PR^2 + QR^2 - 2 * PR * QR * cos(RPQ)
Substituting the given values:
8^2 = (8 * sqrt(3))^2 + 10^2 - 2 * (8 * sqrt(3)) * 10 * cos(RPQ)
Simplifying the equation:
64 = 192 + 100 - 160 * sqrt(3) * cos(RPQ)
Rearranging the equation:
-328 = -160 * sqrt(3) * cos(RPQ)
cos(RPQ) = -328 / (-160 * sqrt(3))
Using an inverse cosine function to find RPQ:
RPQ = acos(-328 / (-160 * sqrt(3)))
RPQ ≈ 149.9 degrees
However, since P is on the north side of Q, increasing the angle of 149.9 degrees by 180 degrees gives us the final bearing of P from R:
Bearing of P from R ≈ 149.9 degrees + 180 degrees ≈ 329.9 degrees
Therefore, the bearing of P from R is approximately 329.9 degrees.