Solution: The volume of a hemisphere is $ \frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi 27x^3 = 18\pi x^3 $. The cylinder's volume is $ \pi x^2 \cdot 3x = 3\pi x^3 $. The ratio is $ \frac{18\pi x^3}{3\pi x^3} = \boxed{6} $.
["Understanding the Volume Ratio: Hemispheres and Cylinders – A Clear Mathematical Solution", "When studying three-dimensional geometry, one common problem involves comparing the volumes of a hemisphere and a cylinder with carefully defined proportions. This scenario not only enhances problem-solving skills but also reveals elegant mathematical relationships that serve as a foundation in calculus and engineering applications.", "### The Hemisphere Volume Formula", "We begin with the volume of a hemisphere, given by the standard formula:", "[\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi r^3\n]", "In this problem, the hemisphere’s radius is specified as ( 3x ). Substituting this into the formula:", "[\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi (3x)^3\n]", "Calculating the cube:", "[\n(3x)^3 = 27x^3\n]", "So,", "[\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi \cdot 27x^3 = 18\pi x^3\n]", "This confirms the hemisphere volume as ( 18\pi x^3 ).", "---", "### The Cylinder Volume Calculation", "The cylinder is defined with radius ( x ) and height ( 3x ). The volume of a cylinder uses the formula:", "[\nV_{\ ext{cylinder}} = \pi r^2 h\n]", "Substituting ( r = x ) and ( h = 3x ):", "[\nV_{\ ext{cylinder}} = \pi (x)^2 \cdot 3x = \pi x^2 \cdot 3x = 3\pi x^3\n]", "Thus, the cylinder’s volume is ( 3\pi x^3 ).", "---", "### Calculating the Volume Ratio", "Now, compute the ratio of the hemisphere’s volume to the cylinder’s volume:", "[\n\ ext{Ratio} = \frac{V_{\ ext{hemisphere}}}{V_{\ ext{cylinder}}} = \frac{18\pi x^3}{3\pi x^3}\n]", "Canceling the common terms ( \pi x^3 ):", "[\n\frac{18}{3} = 6\n]", "So, the volume ratio is exactly 6.", "---", "### Why This Ratio Matters", "This result illustrates a fundamental geometric relationship: the volume of a hemisphere with radius three times that of a cylinder’s base (and height equal to that radius) is six times larger. This proportion is valuable in multiple fields, including physics (fluid displacement), engineering (tank design), and architectural modeling.", "---", "### Key Takeaway", "Knowing how to compute and interpret volumes of basic shapes like hemispheres and cylinders enables students and professionals to solve complex geometry problems efficiently. The calculation:", "[\n\frac{\frac{2}{3}\pi (3x)^3}{\pi x^2 \cdot 3x} = 6\n]", "demonstrates how algebraic manipulation supports clear visual and conceptual understanding—essential for mastering spatial reasoning.", "---", "Summary:\n- Hemisphere volume: ( 18\pi x^3 )\n- Cylinder volume: ( 3\pi x^3 )\n- Volume ratio: ( \boxed{6} )\nThis problem highlights a simple yet powerful volume comparison involving spherical and prism-like shapes."]
