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.
12^2=8^2+10^2-2*8*10cosR
144=64+100-160cosR
144-164=-160cosR
-24/-160=cosR
0.125=cosR
R=cos^-1(0.125)
R= 82.8
The bearing of P from R is 180-30-82.8=67.2
N67E or 067
You sail on a HEADING, not a bearing.
You take a compass bearing on something else as in P from R
Q:
sailed on a heading of 150
which is 30 degrees east of south
(draw that)
we will use the law of cosines
RQ = 10
RP = 8
QP = 12
we want first the angle call it x opposite QP
QP^2 = RQ^2 + RP^2 - 2 QP cos x
144 = 100 + 64 - 24 cos x
cos x = 20/24
x = 33.6 degrees
so the total angle from the south direction east to RP is 30+33.6
= 63.6 east of south
on a compass that bearing is
180 - 63.6 = 116.4 or around ESE
SinR/r=sinQ/q
=sinR/12=sin30/10
cross multiply
10sinR=12sin30
sinR=12sin30/10
R=36.9 or 37
P=R+Q
P=30+37
P=67
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To find the bearing of ship P from port R, we need to use the concept of bearing and apply it to the given information.
First, let's define some variables:
Let PR be the bearing of ship P from port R.
Let QR be the bearing of ship Q from port R.
Let PQ be the bearing of ship P from ship Q.
From the given information:
- Ship Q sailed on a bearing of 150 degrees.
- After a distance of 8 km, ship P is 12 km away from ship Q.
- After a distance of 10 km, ship Q is also 12 km away from ship P.
To start, let's draw a diagram to visualize the scenario:
Q
/ |
/ |
/ |
/ | 12 km
/ |
/ θ |
R-------P
From the diagram, we can see that we have a triangle PRQ.
Now, let's analyze the information step by step:
1. We know that ship Q sailed on a bearing of 150 degrees. This means that the angle QR is 150 degrees.
2. After a distance of 8 km, ship P is 12 km away from ship Q. This gives us the side PQ as 12 km.
3. After a distance of 10 km, ship Q is also 12 km away from ship P. This gives us the side QP as 12 km.
4. We need to find the bearing of ship P from port R, which is angle PRQ.
Now, using the Law of Cosines, we can find angle PRQ:
cos(PRQ) = (PQ^2 + QR^2 - PR^2) / (2 * PQ * QR)
Substituting the given values:
cos(PRQ) = (12^2 + 12^2 - 8^2) / (2 * 12 * 12)
cos(PRQ) = (144 + 144 - 64) / 288
cos(PRQ) = 224 / 288
cos(PRQ) = 0.7777777
To find the angle PRQ, we need to take the inverse cosine (cos^-1) of 0.7777777:
PRQ = cos^-1(0.7777777)
Using a calculator, we find:
PRQ ≈ 41.41 degrees
Therefore, the bearing of ship P from port R is approximately 41.41 degrees.