Final answer:
The question can be solved by converting the bearings to standard angles, determining the change in angle, and then applying the Law of Cosines to calculate the distance between the ship and the submerged rock after it has sailed 16.3 miles.
Step-by-step explanation:
The question involves calculating the distance between a ship and a submerged rock after the ship changes its bearing, using principles from trigonometry and vector subtraction. We're given the initial bearing of the rock from the ship, the path the ship has sailed, and the final bearing of the rock from the ship. To solve this, we need to use the Law of Cosines or vector analysis.
Firstly, we must convert the bearings to standard mathematical angles measured counterclockwise from the positive x-axis (East). The initial bearing of 45°20′ becomes 45.333° (since 20' equals 1/3 of a degree), and the final bearing of 308°40′ becomes 308.667°. Next, we calculate the change in angle by taking the difference, then we apply the Law of Cosines to find the unknown distance from the rock.
Using the Law of Cosines, c² = a² + b² - 2ab * cos(γ), where γ is the angle between the two path vectors, a (the distance initially from the rock) and b (the distance the ship travels, which is 16.3 miles). However, since we typically use degrees and the Law of Cosines requires angles in radians, we need to convert the change in angle to radians before applying the formula.
After calculating the distance, we can provide the answer to the student. The distance of the ship from the rock at the later point can be calculated step by step using the described method.