Solution: The volume of a hemisphere is $ \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (2x)^3 = \frac{16}{3}\pi x^3 $. The cylinder's volume is $ \pi r^2 h = \pi x^2 \cdot 4x = 4\pi x^3 $. The ratio is $ \frac{\frac{16}{3}\pi x^3}{4\pi x^3} = \frac{16}{3} \div 4 = \frac{4}{3} $. Thus, the ratio is $ \boxed{\dfrac{4}{3}} $.

["Solution: Comparing Hemisphere and Cylinder Volumes – How the Ratio Simplifies to $ \frac{4}{3} $", "When exploring geometric volumes, comparing different shapes often reveals surprising relationships and proportional insights. One illuminating example involves finding the volume ratio between a hemisphere and a perfect cylinder — a classic problem in calculus and geometry.", "## The Volumes Explained", "Consider a hemisphere with radius $ r $. Its volume is given by the formula:", "$$\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi r^3\n$$", "Substituting $ r = 2x $, the volume becomes:", "$$\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi (2x)^3 = \frac{2}{3}\pi (8x^3) = \frac{16}{3}\pi x^3\n$$", "Now, compare this to a right circular cylinder with radius $ r = x $ and height $ h = 4x $:", "$$\nV_{\ ext{cylinder}} = \pi r^2 h = \pi x^2 \cdot 4x = 4\pi x^3\n$$", "## Calculating the Volume Ratio", "To find the ratio of the hemisphere’s volume to the cylinder’s volume, divide the two:", "$$\n\ ext{Ratio} = \frac{V_{\ ext{hemisphere}}}{V_{\ ext{cylinder}}} = \frac{\frac{16}{3}\pi x^3}{4\pi x^3}\n$$", "Cancel out $ \pi x^3 $ from numerator and denominator:", "$$\n= \frac{\frac{16}{3}}{4} = \frac{16}{3} \div 4 = \frac{16}{3} \cdot \frac{1}{4} = \frac{16}{12} = \frac{4}{3}\n$$", "## Final Result", "Thus, the volume ratio of the hemisphere to the cylinder is:", "$$\n\boxed{\dfrac{4}{3}}\n$$", "This elegant result shows that the hemisphere occupies $ \frac{4}{3} $ times the volume of the corresponding cylinder when the cylinder’s radius is half that of the hemisphere, and its height matches twice the hemisphere’s radius. Such calculations form the foundation for understanding volume ratios in engineering, architecture, and physics.", "Whether you're designing tanks, analyzing fluid dynamics, or teaching geometry, mastering volume relationships helps simplify complex problems into clear, actionable insights."]