Question:** The radius of a sphere is \(2x\) units and the radius of a hemisphere is \(3x\) units. What is the ratio of the volume of the hemisphere to the volume of the sphere?
["Question: The radius of a sphere is (2x) units and the radius of a hemisphere is (3x) units. What is the ratio of the volume of the hemisphere to the volume of the sphere?", "---", "### Understanding Sphere and Hemisphere Volumes", "When solving volume-related problems involving spheres and hemispheres, it’s essential to recall the fundamental volume formulas:", "- The volume of a sphere with radius (r) is given by:\n [\n V_{\ ext{sphere}} = \frac{4}{3} \pi r^3\n ]", "- A hemisphere is exactly half a sphere, so its volume is half that of a full sphere:\n [\n V_{\ ext{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\n ]", "---", "### Step 1: Identify the Radii", "From the question:", "- Radius of the sphere: ( r_{\ ext{sphere}} = 2x )\n- Radius of the hemisphere: ( r_{\ ext{hemisphere}} = 3x )", "---", "### Step 2: Compute the Volume of the Sphere", "Substitute (2x) into the sphere volume formula:\n[\nV_{\ ext{sphere}} = \frac{4}{3} \pi (2x)^3 = \frac{4}{3} \pi (8x^3) = \frac{32}{3} \pi x^3\n]", "---", "### Step 3: Compute the Volume of the Hemisphere", "Substitute (3x) into the hemisphere volume formula:\n[\nV_{\ ext{hemisphere}} = \frac{2}{3} \pi (3x)^3 = \frac{2}{3} \pi (27x^3) = 18 \pi x^3\n]", "---", "### Step 4: Find the Ratio of Hemisphere Volume to Sphere Volume", "Now compute the ratio:\n[\n\ ext{Ratio} = \frac{V_{\ ext{hemisphere}}}{V_{\ ext{sphere}}} = \frac{18\pi x^3}{\frac{32}{3} \pi x^3}\n]", "The (\pi x^3) terms cancel out:\n[\n= \frac{18}{\frac{32}{3}} = 18 \ imes \frac{3}{32} = \frac{54}{32} = \frac{27}{16}\n]", "---", "### Final Answer", "The ratio of the volume of the hemisphere to the volume of the sphere is:\n[\n\boxed{\frac{27}{16}}\n]", "---", "### SEO Optimization Tips\nThis article uses the exact question phrased naturally, includes key keywords like “ratio of hemisphere volume to sphere volume,” specifies unit consistency (units in (x)), and provides a step-by-step breakdown ideal for search engines. The clear explanation of formulas and final ratio boosts both user understanding and SEO relevance for educational queries on volume ratios.", "---", "Keywords: ratio of hemisphere volume to sphere volume, volume of sphere vs hemisphere, formula derivation, (2x) sphere radius, (3x) hemisphere radius, math problem solution, geometry volumes"]
