Question: A space policy analyst evaluates a spherical asteroid with radius $ r $ and a cylindrical asteroid miner with the same volume. If the cylinders height equals its diameter, what is the ratio of the cylinders height to the spheres radius?

["How Space Mining Design Balances Form and Function – The Asteroid Miner Riddle", "In a growing wave of interest around off-world resource utilization, a fascinating engineering question has emerged: if a spherical asteroid of radius $ r $ is evaluated alongside a specialized cylindrical miner that matches its volume and maintains a height equal to its diameter, what is the ratio of the cylinder’s height to the sphere’s radius? This isn’t just a technical curiosity—it touches on smart design in extraterrestrial resource extraction, drawing attention from space policy analysts and industry planners across the U.S.", "As global investment in sustainable space technology accelerates, efficient use of asteroid materials has become essential. Spherical shapes offer structural advantage and minimal surface exposure, but cylindrical mining structures provide practical storage and processing volume. When designed to match volume while maintaining a height equal to diameter, the geometry creates a measurable engineering invariant—one that reveals deeper insights into volume efficiency and spatial optimization.", "Understanding the Volumes: Sphere versus Cylinder \nThe volume of a sphere is $ V_{\ ext{sphere}} = \frac{4}{3}\pi r^3 $. \nFor the cylinder, volume is $ V_{\ ext{cylinder}} = \pi R^2 h $, where $ R $ is the radius and $ h $ is the height. \nThe condition $ h = 2R $ transforms the cylinder’s volume into $ V_{\ ext{cylinder}} = \pi R^2 (2R) = 2\pi R^3 $. \nSet volumes equal: \n$$\n\frac{4}{3}\pi r^3 = 2\pi R^3\n$$ \nCancel $ \pi $ and solve: \n$$\n\frac{4}{3} r^3 = 2 R^3 \Rightarrow R^3"]