Question:** A cylinder has a height equal to twice its radius. If the height is increased by 50% and the radius is decreased by 20%, what is the ratio of the new volume to the original volume?
["Question:\nA cylinder has a height equal to twice its radius. If the height is increased by 50% and the radius is decreased by 20%, what is the ratio of the new volume to the original volume?", "---", "### Understanding Cylinder Volume Before and After Changes", "When analyzing geometric transformations like changes in dimensions, volume calculations remain key—especially for shapes like cylinders defined by the formula:\n[ V = \pi r^2 h ]\nwhere ( r ) is the radius and ( h ) is the height.", "In this problem, the cylinder starts with a height exactly twice the radius, so let’s define the original dimensions using a variable:", "- Let radius ( r = r )\n- Then original height ( h = 2r )", "Using this, the original volume ( V_{\ ext{original}} ) is:\n[\nV_{\ ext{original}} = \pi r^2 (2r) = 2\pi r^3\n]", "---", "### Applying the Changes", "Next, we apply the specified modifications:", "- Height increased by 50%:\n New height:\n [\n h_{\ ext{new}} = 2r + 0.5(2r) = 3r\n ]", "- Radius decreased by 20%:\n New radius:\n [\n r_{\ ext{new}} = r - 0.2r = 0.8r\n ]", "---", "### Calculate the New Volume", "Substitute the updated dimensions into the volume formula:\n[\nV_{\ ext{new}} = \pi (r_{\ ext{new}})^2 \cdot h_{\ ext{new}} = \pi (0.8r)^2 (3r)\n]", "Simplify step-by-step:\n[\n= \pi (0.64r^2)(3r) = \pi \cdot 1.92r^3 = 1.92\pi r^3\n]", "---", "### Find the Volume Ratio (New to Original)", "Now compute the ratio:\n[\n\frac{V_{\ ext{new}}}{V_{\ ext{original}}} = \frac{1.92\pi r^3}{2\pi r^3} = \frac{1.92}{2} = 0.96\n]", "---", "### Final Answer and Explanation", "The ratio of the new volume to the original volume is:\n[\n\boxed{0.96}\n]\nor equivalently, 96% of the original volume — meaning the volume decreased by 4% after the applied changes.", "This result highlights how proportional changes in dimensions affect three-dimensional space: halving the height and reducing the radius both reduce total volume, but combined in this case results in a predictable, multiplicative reduction.", "---", "Keywords for SEO optimization:\nCylinder volume ratio, geometric shape transformation, volume formula cylinder, changing cylinder dimensions, ratio of volumes, height increased 50 cylinder, radius decreased 20%, cylinder volume change, mathematical ratio calculation, cylinder math ratio, radius and height change volume"]
