Final answer:
The length of the arc of a great circle on Earth subtended by an angle of 1 minute at the center is approximately 1.853 kilometers.
Step-by-step explanation:
The question relates to the computation of the arc length of a great circle on Earth that subtends a central angle of 1 minute of arc (or 1/60 of a degree). To find the arc length, we need to use the formula for the circumference of the Earth and then calculate the proportion of that circumference that corresponds to the angle in question. The circumference of the Earth at the equator (which is a great circle) is approximately 40,075 kilometers (the exact value may vary slightly depending on the source).
Since there are 360 degrees in a full revolution and each degree is divided into 60 minutes, the proportion of the Earth's circumference that corresponds to 1 minute of arc is (1/60) / 360, or 1/21600 of the full circumference.
Therefore, we calculate the arc length as follows:
- Find the fraction of the circumference one minute of arc represents: 1 minute = 1/60 degree, so the fraction is 1/60 ÷ 360 = 1/21600.
- Multiply this fraction by the Earth's circumference: Arc length = Circumference × (1/21600).
- Substitute the approximate value for the Earth's circumference at the equator: Arc length ≈ 40,075 km × (1/21600) = 1.853 kilometers.
Thus, the length of the arc of a great circle on Earth subtended by an angle of 1 minute at the center is approximately 1.853 kilometers (or 1853 meters).