Question

A ship leaves port at noon and has a bearing of S29oW. The ship sails at 20 knots. How many nautical miles south and how many nautical miles west will the ship have traveled by 6:00 PM

Answer 1

Approximately
`58.2\; \text{nautical miles}` (assuming that the bearing is
`{\rm S$29^(\circ)$W}`.)

Explanation:

Let
`v` denote the speed of the ship, and let
`t` denote the duration of the trip. The magnitude of the displacement of this ship would be
`v\, t`.

Refer to the diagram attached. The direction
`{\rm S$29^(\circ)$W}` means
`29^(\circ)` west of south. Thus, start with the south direction and turn towards west (clockwise) by
`29^(\circ)` to find the direction of the displacement of the ship.

The hypothenuse of the right triangle in this diagram represents the displacement of the ship, with a length of
`v\, t`. The dashed horizontal line segment represents the distance that the ship has travelled to the west (which this question is asking for.) The angle opposite to that line segment is exactly
`29^(\circ)`.

Since the hypotenuse is of length
`v\, t`, the dashed line segment opposite to the
`\theta = 29^(\circ)` vertex would have a length of:

`\begin{aligned}& \text{opposite (to $\theta$)} \\ =\; & \text{hypotenuse} * \frac{\text{opposite (to $\theta$)}}{\text{hypotenuse}} \\ =\; & \text{hypotenuse} * \sin (\theta) \\ =\; & v\, t \, \sin(\theta) \\ =\; & v\, t\, \sin(29^(\circ))\end{aligned}`.

Substitute in
`\begin{aligned} v &= 20\; \frac{\text{nautical mile}}{\text{hour}}\end{aligned}` and
`t = 6\; \text{hour}`:

`\begin{aligned} & v\, t\, \sin(29^(\circ)) \\ =\; & 20\; \frac{\text{nautical mile}}{\text{hour}} * 6\; \text{hour} * \sin(29^(\circ)) \\ \approx\; & 58.2\; \text{nautical mile}\end{aligned}`.