The initial course of a certain ship sailing from San Francisco (37o46’N, 122o25’W) is due west. After she has sailed 1000 nautical miles on a great circle, find her: Task 1 – Latitude

Answer 1

Let's approach this step-by-step to find the latitude of the ship after sailing 1000 nautical miles due west from San Francisco on a great circle.

1) First, let's clarify the given information:

- Starting point: San Francisco (37°46'N, 122°25'W)

- Initial course: Due west

- Distance sailed: 1000 nautical miles on a great circle

2) For a course due west, the latitude will remain constant on a rhumb line. However, on a great circle, the latitude will change slightly, moving southward in the Northern Hemisphere.

3) To solve this, we'll use the formula for latitude (φ) after sailing a distance (d) on an initial course of 90° (due west):

sin(φ) = sin(φ₁) * cos(d/R)

Where:

φ₁ is the initial latitude

d is the distance sailed in nautical miles

R is the Earth's radius in nautical miles (approximately 3440 nautical miles)

4) Let's convert our initial latitude to decimal degrees:

37°46'N = 37 + (46/60) = 37.7667°

5) Now, let's plug our values into the formula:

sin(φ) = sin(37.7667°) * cos(1000/3440)

6) Calculating:

sin(φ) = 0.6120 * 0.9553 = 0.5846

7) To get φ, we need to take the inverse sine (arcsin):

φ = arcsin(0.5846) = 35.8048°

8) Converting back to degrees and minutes:

35.8048° = 35°48.29'N

Therefore, after sailing 1000 nautical miles due west on a great circle from San Francisco, the ship's latitude will be approximately 35°48'N.