P,Q,R,S are four locations the same horizontal plane Q is on bearing of 41 degrees from R and the distance is 40km. S is 520km from R on a bearing 074 degrees directly due north of P and the distance between Q and R is 38km . Find to the nearest whole numbers
a) the bearing R from Q
b) the distance between Q and S
C) the distance between P and R
a) To find the bearing of R from Q, we need to find the angle QRS.
Angle QRS = 180 degrees - Angle PQR - Angle PQS
Angle PQS = 90 degrees - Angle SPQ (because SP is directly north of PQ)
Angle SPQ = 180 degrees - Angle QPS - Angle PSQ
Since Angle PSQ is given as 074 degrees, we can find Angle QPS using the triangle QPS:
Angle QPS = 180 degrees - Angle PSQ - Angle PQS
= 180 degrees - 074 degrees - 90 degrees
= 16 degrees
And Angle PQR = 180 degrees - Angle QRP - Angle RPQ
= 180 degrees - 041 degrees - 90 degrees
= 49 degrees
Plugging in these values, we have:
Angle PQS = 16 degrees
Angle PQR = 49 degrees
Angle QRS = 180 degrees - 49 degrees - 16 degrees
= 115 degrees
Therefore, the bearing R from Q is 115 degrees.
b) To find the distance between Q and S, we can use the law of cosines in triangle QSR:
QS^2 = QR^2 + SR^2 - 2(QR)(SR)cos(QRS)
QS^2 = 40^2 + 520^2 - 2(40)(520)cos(115 degrees)
= 1600 + 270400 - 41600cos(115 degrees)
Using a calculator to find cos(115 degrees), we get:
QS^2 = 272000 + 41600(0.42262)
= 272000 + 17563.52
= 289563.52
Taking the square root, we get:
QS ≈ 538 km
Therefore, the distance between Q and S is approximately 538 km.
c) To find the distance between P and R, we can use the Pythagorean theorem in right triangle PQS:
PR^2 = PQ^2 + QS^2
PR^2 = 38^2 + 538^2
PR^2 = 1444 + 289444
PR^2 = 290888
Taking the square root, we get:
PR ≈ 539 km
Therefore, the distance between P and R is approximately 539 km.