Part a: Calculate the Difference in Distance

To determine the distance each vessel will sail following their shortest legal route, we'll need to use the Great Circle distance between the departure and arrival points. However, since the vessels are overloaded, they may be restricted to staying below the Winter Load Line (35°N), implying that they might have to deviate from the Great Circle route. We will calculate two routes:

1. Great Circle Route: The shortest path over the surface of a sphere, using the haversine formula for distance between two points.
2. Orthodromic Route with Latitude Restriction: The shortest path that involves reaching 35°N and then traveling along a rhumb line (constant latitude) to the destination.

Coordinates:

• Departure: $20^\circ 27'N$, $121^\circ 58'E$
• Arrival: $37^\circ 45'N$, $122^\circ 42'W$

We'll calculate both distances and compare them.

1. Great Circle Distance Formula:

The formula to calculate the distance $d$ between two points $(\phi_1, \lambda_1)$ and $(\phi_2, \lambda_2)$ on a sphere is:

$d = R \cdot \arccos \left( \sin(\phi_1) \sin(\phi_2) + \cos(\phi_1) \cos(\phi_2) \cos(\Delta \lambda) \right)$

where:

• $\phi_1$ and $\phi_2$ are the latitudes of the two points (in radians),
• $\lambda_1$ and $\lambda_2$ are the longitudes of the two points (in radians),
• $\Delta \lambda = \lambda_2 - \lambda_1$ is the difference in longitude,
• $R$ is the Earth's radius (mean radius = 6,371 km).

We'll first calculate the Great Circle distance between the departure and arrival points. Then we'll calculate the distance via 35°N to compare the two.

Let me calculate that.