Solution: The volume of a sphere of radius $ x $ is $ V_{ ext{particle}} = rac{4}{3}\pi x^3 $. The volume of a hemisphere of radius $ 3x $ is half the volume of a full sphere:
["Understanding the Volume of a Sphere and Hemisphere: A Clear Mathematical Solution", "When studying geometric volumes, one essential formula defines the space occupied by round objects in three-dimensional space—such as particles or celestial bodies. The volume of a sphere with radius $ x $ is accurately given by:", "$$\nV_{\ ext{particle}} = \frac{4}{3}\pi x^3\n$$", "This formula is fundamental in physics, engineering, and applied mathematics, especially when analyzing particle distribution, fluid dynamics, or gravitational models.", "Now, consider a hemisphere—half of a full sphere—with radius $ 3x $. Since a hemisphere constitutes precisely half the volume of a complete sphere, we compute:", "First, calculate the volume of a full sphere of radius $ 3x $ using the standard formula:", "$$\nV_{\ ext{sphere}} = \frac{4}{3}\pi (3x)^3 = \frac{4}{3}\pi (27x^3) = 36\pi x^3\n$$", "Since the hemisphere is half of this:", "$$\nV_{\ ext{hemisphere}} = \frac{1}{2} \ imes 36\pi x^3 = 18\pi x^3\n$$", "Thus, the volume of a hemisphere of radius $ 3x $ is $ 18\pi x^3 $. This precise derivation highlights the importance of understanding how scaling a radius by a factor affects volume—specifically, volume scales with the cube of the radius, meaning tripling the radius increases volume by $ 3^3 = 27$.", "Whether modeling planetary bodies, analyzing spherical particles in nanotechnology, or simulating physical systems, knowing these volume relationships ensures accurate predictions and efficient problem-solving.", "---", "Key Takeaways:", "- Volume of a sphere of radius $ x $: $ V = \frac{4}{3}\pi x^3 $\n- Hemisphere of radius $ 3x $: $ V = 18\pi x^3 $\n- Radius scaling impacts volume cubically — key for dimensional analysis and real-world modeling", "Understanding these geometric principles enables deeper insights in fields ranging from materials science to astrophysics."]
