Solution: The total radius of the orbit is the sum of the planet's radius and the satellite's orbital radius: $ 6,000 + 10,000 = 16,000 $ km. The circumference $ C $ is calculated as $ C = 2\pi \times 16,000 = 32,000\pi $ km.
["Understanding Satellite Orbits: Calculating Circumference Using Total Radius", "When studying satellite motion, one fundamental calculation is the satellite’s orbital circumference — the distance a satellite travels in a full orbit. A clear solution approach involves determining the total orbital radius first and then applying the circle formula for circumference.", "### The Simple Mathematical Solution", "The total radius of a satellite’s orbit is found by adding the radius of the planet to the satellite’s orbital altitude. According to the given data:", "- Planet’s radius: 6,000 km\n- Satellite’s orbital radius (altitude): 10,000 km", "Thus, the total orbital radius is:\n[ 6,000\ \ ext{km} + 10,000\ \ ext{km} = 16,000\ \ ext{km} ]", "With this total radius, the circumference $ C $ of the orbit is calculated using the standard formula for the circumference of a circle:\n[ C = 2\pi r ]\nSubstituting $ r = 16,000 $ km:\n[ C = 2\pi \ imes 16,000 = 32,000\pi\ \ ext{km} ]", "### Why This Matters", "Calculating orbital circumference is vital for satellite mission planning, communication coverage analysis, and predicting orbital periods. Knowing the total path a satellite travels allows engineers to estimate signal travel times, fuel requirements, and coverage areas.", "### Why the Total Radius Addition Works", "Adding the planet’s radius to the satellite’s orbital altitude reflects the distance from the planet’s center to the farthest point of the orbit. This total radial distance defines the radius around which the satellite travels entirely.", "### Final Result", "For a satellite orbiting at 10,000 km above a planet with a 6,000 km radius, the full orbital circumference is:\n[ C = 32,000\pi\ \ ext{km} ]\nApproximately 100,530 km (using $ \pi \approx 3.1416 $).", "---", "This mathematical approach provides a clear, efficient way to understand a key satellite orbital parameter — combining planetary and orbital data to uncover the complete journey distance traveled in orbit."]
