Final answer:
The yacht is located approximately 4.09 nautical miles east and 14.53 nautical miles north of Port Latta, Tasmania. The new course the yacht needs to follow to reach Port Latta is approximately 74.21° (measured clockwise from due north).
Step-by-step explanation:
The yacht started at Port Latta, Tasmania and followed a course of 075° (measured clockwise from due north) for 15 nautical miles. To identify the location of point X, we need to find the end point of this course. We can use trigonometry to determine the north and east components traveled by the yacht.
For the north component, we can use sine: sin(75°) = N/15, where N represents the north component. Solving for N, we get N = 15 * sin(75°) ≈ 14.53 nautical miles.
For the east component, we can use cosine: cos(75°) = E/15, where E represents the east component. Solving for E, we get E = 15 * cos(75°) ≈ 4.09 nautical miles.
Therefore, the location of point X is approximately 4.09 nautical miles east and 14.53 nautical miles north of Port Latta, Tasmania.
To determine the new course the yacht needs to follow to reach Port Latta, we can use trigonometry again. We need to find the angle at point X that is formed between the north direction and the line connecting point X and Port Latta.
Using tangent, we can find the angle: tan(A) = N/E, where A represents the angle. Solving for A, we get A = atan(N/E) = atan(14.53/4.09) ≈ 74.21°.
Therefore, the new course the yacht needs to follow to reach Port Latta is approximately 74.21° (measured clockwise from due north).