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This problem is about calculating the latitude of a vessel based on a given distance traveled due east and the change in longitude. The formula that relates distance, change in longitude, and latitude for a journey along a parallel of latitude on a spherical Earth is:

\[
d = \Delta \lambda \cdot \cos(\phi) \cdot \frac{111.32 \, \text{km}}{1^\circ}
\]

Where:
- \(d\) is the distance traveled eastward (in nautical miles).
- \(\Delta \lambda\) is the change in longitude in degrees.
- \(\phi\) is the latitude (which we are solving for).
- 1 degree of longitude is approximately 111.32 km at the equator (we'll convert to nautical miles).
- 1 nautical mile = 1.852 km.

### Given:
- Distance \(d = 245 \, \text{NM}\)
- Change in longitude \(\Delta \lambda = 7^\circ 25' = 7 + \frac{25}{60} = 7.4167^\circ\)

### Step-by-Step Calculation:

1. Convert the distance into kilometers:
   \[
   245 \, \text{NM} = 245 \times 1.852 = 453.74 \, \text{km}
   \]

2. Now use the formula and rearrange to solve for \(\cos(\phi)\):
   \[
   453.74 = 7.4167 \cdot \cos(\phi) \cdot 111.32
   \]

3. Simplify and solve for \(\cos(\phi)\):
   \[
   \cos(\phi) = \frac{453.74}{7.4167 \cdot 111.32} = \frac{453.74}{825.79} = 0.5494
   \]

4. Find \(\phi\):
   \[
   \phi = \cos^{-1}(0.5494) \approx 56.74^\circ
   \]

### Answer:
The vessel was approximately at a latitude of \( 56.74^\circ \) North (or South depending on the direction).

Would you like any further clarification or details on this calculation?

Here are 5 related questions:
1. How would the formula change if the vessel was traveling along a meridian (north-south)?
2. What is the significance of using nautical miles in navigational problems?
3. How would a change in the Earth's radius affect the calculation of latitude?
4. Can this method be applied to vessels moving westward?
5. How does the curvature of the Earth influence long-distance navigational calculations?

**Tip:** Always ensure that the units used (nautical miles, kilometers, degrees) are consistent throughout calculations to avoid errors.

By sailing due east for a distance of 245 NM, a vessel changes its longitude by 7° 25'. What latitude was the vessel in?

Learn how to calculate the latitude of a vessel by solving a problem involving eastward travel of 245 nautical miles and a longitude change of 7° 25'. This problem involves trigonometric concepts and spherical geometry, suitable for Grades 10-12 or college-level students.

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