1)	2 Gas Carriers are to sail from Basco (Philippines) to San Francisco (California).
One is 540 tonnes and the other 750 tonnes overloaded against the Winter Load Line (35°N).  

Each vessel consumes 95 tonnes of fuel and 12 tonnes of water per day and travels at 22.5 knots.

Departure position off the Basco: 	20° 27’N 121° 58’E
Arrival position off San Francisco: 	37° 45’N 122° 42’W

a)	Calculate the difference in distance the two vessels will sail if they each follow their shortest legal route.




b)	Also calculate the Longitude the direct Great Circle crosses the Latitude of 35°N.



### 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.

2 Gas Carriers are to sail from Basco (Philippines) to San Francisco (California). One is 540 tonnes and the other 750 tonnes overloaded against the Winter Load Line (35°N). Each vessel consumes 95 tonnes of fuel and 12 tonnes of water per day and travels at 22.5 knots. Departure position off the Basco: 20° 27’N 121° 58’E Arrival position off San Francisco: 37° 45’N 122° 42’W a) Calculate the difference in distance the two vessels will sail if they each follow their shortest legal route. b) Also calculate the Longitude the direct Great Circle crosses the Latitude of 35°N.

Learn how to calculate the Great Circle distance and Rhumb Line for two overloaded gas carriers sailing from Basco (Philippines) to San Francisco (California). This detailed solution covers spherical trigonometry concepts, including Great Circle distance formulas, the haversine formula, and the effect of latitude restrictions due to the Winter Load Line. Ideal for advanced navigation and maritime studies.

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