Question: A cylinder has radius $ r $ and height $ 6r $. A cone has radius $ 2r $ and height $ 3r $. What is the ratio of the volume of the cylinder to the volume of the cone?
["Title: Comparing Volumes: The Cylinder vs. The Cone in Geometry", "When studying three-dimensional shapes, one key concept is volume — the amount of space inside a solid. Two common solids are the cylinder and the cone, each defined by unique dimensions. Understanding their volume ratios helps in solving real-world problems in engineering, architecture, and physics.", "In this article, we’ll explore the volumes of a cylinder with radius $ r $ and height $ 6r $, and a cone with radius $ 2r $ and height $ 3r $. By calculating the volume of each and forming their ratio, we reveal how these geometric figures compare in capacity.", "---", "### Cylinder Volume Formula", "The volume $ V_{\ ext{cyl}} $ of a cylinder is given by:", "[\nV_{\ ext{cyl}} = \pi r^2 h\n]", "Where:\n- $ r $ is the radius\n- $ h $ is the height", "For our cylinder:\n- Radius = $ r $\n- Height = $ 6r $", "Substituting into the formula:", "[\nV_{\ ext{cyl}} = \pi r^2 (6r) = 6\pi r^3\n]", "---", "### Cone Volume Formula", "The volume $ V_{\ ext{cone}} $ of a cone is:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi r^2 h\n]", "Where:\n- $ r $ is the radius\n- $ h $ is the height", "For our cone:\n- Radius = $ 2r $\n- Height = $ 3r $", "Substituting:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi (2r)^2 (3r) = \frac{1}{3} \pi (4r^2)(3r) = \frac{1}{3} \pi \cdot 12r^3 = 4\pi r^3\n]", "---", "### Ratio of Volumes", "Now, compute the ratio of the volume of the cylinder to the volume of the cone:", "[\n\ ext{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 ratio of the volume of the cylinder to the volume of the cone is:", "[\n\boxed{\frac{3}{2}}\n]", "---", "### Interpretation", "This means the cylinder holds 1.5 times the volume of the cone — a crucial insight when comparing storage capacity, structural design, or fluid displacement in cylindrical and conical containers.", "Understanding such ratios enhances problem-solving in mathematics and applied sciences, providing clarity on spatial relationships between geometric forms.", "---", "Keywords for SEO:\ncylinder volume vs cone volume, cylinder height 6r, cone radius 2r, volume ratio cylinder cone, geometric volume comparison, math volume formula, cylinder to cone ratio, geometry basics, cylinder volume formula, cone volume calculation", "Meta Description:\nCompare the volume of a cylinder (radius $ r $, height $ 6r $) to a cone (radius $ 2r $, height $ 3r $). Find the ratio of cylinder volume to cone volume: $ \frac{3}{2} $, meaning the cylinder holds 1.5 times the cone’s volume. Ideal for math students and geometry applications."]
