Solution: The volume of the cylinder is $ V_{\text{cyl}} = \pi r^2 (6r) = 6\pi r^3 $. The volume of the cone is $ V_{\text{cone}} = \frac{1}{3} \pi (2r)^2 (3r) = \frac{1}{3} \pi (4r^2)(3r) = 4\pi r^3 $. The ratio is:
["Understanding the Volume Ratio: Cylinder to Cone with Related Dimensions", "When exploring geometric solids, one of the most insightful problems involves analyzing the volume ratio between different shapes — particularly cylinders and cones with proportional dimensions. This article delves into a classic volume comparison: a cylinder with radius ( r ) and height ( 6r ), versus a cone with radius ( 2r ) and height ( 3r ). We derive their respective volumes, reveal the volume ratio, and highlight the practical significance of this relationship in real-world applications.", "---", "### Cylinder Volume Calculated", "The volume of a cylinder is given by the formula:", "[\nV_{\ ext{cyl}} = \pi r^2 h\n]", "Substituting the given height ( h = 6r ):", "[\nV_{\ ext{cyl}} = \pi r^2 (6r) = 6\pi r^3\n]", "---", "### Cone Volume Computed", "The volume of a cone follows:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi r^2 h\n]", "Here, the radius is ( 2r ) and the height is ( 3r ), so:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi (2r)^2 (3r) = \frac{1}{3} \pi (4r^2)(3r) = \frac{1}{3} \ imes 12\pi r^3 = 4\pi r^3\n]", "---", "### Volume Ratio: Cylinder to Cone", "The key ratio of interest is:", "[\n\ ext{Volume Ratio} = \frac{V_{\ ext{cyl}}}{V_{\ ext{cone}}} = \frac{6\pi r^3}{4\pi r^3} = \frac{6}{4} = \frac{3}{2}\n]", "Thus, the volume of the cylinder is ( \frac{3}{2} ) times (or 1.5 times) the volume of the cone.", "---", "### Why This Ratio Matters: Geometric Consistency and Applications", "The constant ratio of ( 3:2 ) arises naturally from the proportional scaling of dimensions — doubling the radius and tripling the height of a cone relative to a cylinder (relative to base radius scaling) leads to this fixed volume relationship. This geometric consistency makes such comparisons essential in engineering, architecture, and manufacturing — for instance, when estimating material volumes or optimizing container designs.", "Despite differing amounts, the cylinder holds nearly 1.5× the volume of the cone, which informs decisions about storage space, load capacity, or production efficiency where such shapes are involved.", "---", "Summary\nGiven a cylinder of radius ( r ) and height ( 6r ), and a cone of radius ( 2r ) and height ( 3r ), the volume ratio is:", "[\n\boxed{\frac{V_{\ ext{cyl}}}{V_{\ ext{cone}}} = \frac{3}{2}}\n]", "This result underscores a fundamental geometric proportion and aids in practical problem-solving across multiple disciplines."]
