The bearing to take is 134 degrees and the shortest distance is 14.3 km
The bearing it would take
The attached diagram represents a labelled version of the ship movement
The angle at Q is
Q = 180 - 155 + 90 + 120 - 90
This gives
Q = 145
The distance PR is calculated as:
PR^2 =PQ^2 + QR^2 - 2 * PQ * QR * cos(Q)
This gives
PR^2 = 9^2 + 6^2 - 2 * 9 * 6 * cos(145)
Evaluate
PR^2 = 205.46
Take the square root
PR = 14.3 km
The angle at P is then calculated as:
QR/sin(P) = PR/sin(Q)
This gives
6/sin(P) = 14.3/sin(145)
Evaluate the quotient
6/sin(P) = 24.93
Make sin(P) the subject
sin(P) = 6/24.93
Divide
sin(P) = 0.2407
Take the arc sin of both sides
P = 14
The bearing is then calculated as:
Bearing = P + 120
This gives
Bearing = 14 + 120
Evaluate
Bearing = 134
Hence, the bearing to take is 134 degrees
The shortest distance
In (a), we have
PR = 14.3 km
This represents the shortest distance
Hence, the shortest distance is 14.3 km
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