A ship leaves POrt P and sails 28km away on bearing of 70 degrees to POrt Q. The ship then turns on a bearing of 160 degrees and sails 21km to POrt R. Find angle PQR, calculate the distance PR, Determine the bearing of R from P.
angle PQR = 180 degrees - 160 degrees = 20 degrees
ships sail on headings, not bearings.
Angle PQR is not the angle between the NS line and R. It is the angle between PQ and QR. That is, 70 + 20 = 90 degrees.
So, now we can figure the x-y displacements:
PQ = <28sin70°,28cos70°> = <26.31,9.58>
QR = <21sin20°,-21cos20°> = <7.18,-19.73>
Add them up and you get PR = <33.49,-10.15> = 35.00 at 106.88°
Dp yygg
To calculate the distance PR, we can use the Law of Cosines. In triangle PQR, we have the side lengths PQ = 28km, QR = 21km, and the angle PQR = 20 degrees.
Using the Law of Cosines formula, we have:
PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)cos(PQR)
PR^2 = (28km)^2 + (21km)^2 - 2(28km)(21km)cos(20 degrees)
PR^2 = 784km^2 + 441km^2 - 1176km^2cos(20 degrees)
PR^2 = 1225km^2 - 1176km^2cos(20 degrees)
Now, we can use a calculator to find the value of cos(20 degrees) and substitute it into the equation to find PR^2. Finally, take the square root of PR^2 to find the distance PR.
To determine the bearing of R from P, we need to calculate the angle PRP. Since we already know angle PQR is 20 degrees, we can find angle PRP using the equation:
angle PRP = 180 degrees - angle PQR
Substitute the value of angle PQR (20 degrees) into the equation:
angle PRP = 180 degrees - 20 degrees
angle PRP = 160 degrees
Therefore, the bearing of R from P is 160 degrees.