Final answer:
To maintain a true course due west while encountering a 3.0-knot current running from north to south, the ship should have a heading of 450° (measured clockwise from the north). It will take the ship 1.25 hours to proceed 20 nautical miles due west.
Step-by-step explanation:
When a ship encounters a current, the heading of the ship needs to be adjusted to compensate for the current and maintain the desired course. To determine the heading θ of the ship, we can use vector addition.
In this scenario, the ship is heading due west (270°) and the current is running from north to south (180°). To maintain the true course due west, the ship needs to have a resultant velocity equal to the speed of the ship in still water (16 knots) and in the westward direction. This means that the ship needs to have a heading of θ = 270° + 180° = 450° (measured clockwise from the north).
To calculate the time it takes for the ship to proceed 20 nautical miles due west, we can use the formula:
$t = \frac{d}{v}$
Where t is the time, d is the distance, and v is the velocity. In this case, the distance is 20 nautical miles and the velocity is the speed of the ship in still water (16 knots). Plugging in the values, we get:
$t = \frac{20}{16} = 1.25$ hours