A cone with a radius of 4 cm and a height of 9 cm is melted to form a sphere. What is the radius of the sphere?
["Title: How to Calculate the Radius of a Sphere Formed from a Cone: A Step-by-Step Guide", "When a physical shape like a cone is melted and reshaped into another form—such as a sphere—key geometric properties remain conserved, especially volume. In this article, we explore how to determine the radius of a sphere formed from a cone, using precise mathematical principles. Specifically, we analyze a cone with a radius of 4 cm and a height of 9 cm that is fully melted and recast into a perfect sphere.", "---", "### Why Volume Matters When Reshaping Solids", "Melting and reshaping objects preserves volume. Since the cone is melted into a sphere, both shapes have the same volume. By equating the volume formula of the cone to that of the sphere, we can find the sphere’s radius.", "---", "### Step 1: Volume of the Original Cone", "The formula for the volume ( V ) of a cone is:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi r^2 h\n]", "Given:\n- Radius ( r = 4 ) cm\n- Height ( h = 9 ) cm", "Calculate:", "[\nV_{\ ext{cone}} = \frac{1}{3} \pi (4)^2 (9) = \frac{1}{3} \pi \cdot 16 \cdot 9 = \frac{144}{3} \pi = 48\pi \ ext{ cm}^3\n]", "So, the volume of the cone is ( 48\pi ) cubic centimeters.", "---", "### Step 2: Volume of the Sphere", "The volume formula for a sphere is:", "[\nV_{\ ext{sphere}} = \frac{4}{3} \pi R^3\n]", "where ( R ) is the radius of the sphere we need to find.", "Since the volume remains constant during melting:", "[\nV_{\ ext{sphere}} = V_{\ ext{cone}} = 48\pi\n]", "So:", "[\n\frac{4}{3} \pi R^3 = 48\pi\n]", "---", "### Step 3: Solve for ( R )", "First, divide both sides by ( \pi ):", "[\n\frac{4}{3} R^3 = 48\n]", "Next, multiply both sides by ( \frac{3}{4} ):", "[\nR^3 = 48 \ imes \frac{3}{4} = 36\n]", "Now take the cube root:", "[\nR = \sqrt[3]{36}\n]", "---", "### Step 4: Approximate the Radius", "The cube root of 36 is not a whole number. Approximating:", "[\nR \approx 3.30 \ ext{ cm}\n]", "(More precisely: ( \sqrt[3]{36} \approx 3.3019 ), which rounds to 3.30 cm.)", "---", "### Conclusion: Radius of the Sphere", "When a cone with a radius of 4 cm and a height of 9 cm is melted and recast into a sphere, the resulting sphere has a radius of approximately 3.30 cm.", "This example demonstrates how volume conservation bridges different geometric forms and allows precise calculations in practical and theoretical geometry. Whether for designs, manufacturing, or mathematical study, understanding shape transformations ensures accuracy and efficiency.", "---", "### Key Takeaways:", "- Cone volume: ( V = \dfrac{1}{3} \pi r^2 h )\n- Sphere volume: ( V = \dfrac{4}{3} \pi R^3 )\n- Equating volumes allows solving for new shape radius after melting/reshaping\n- Numerically: for cone (r=4, h=9), melting forms sphere with radius ≈ 3.30 cm", "Understanding these principles helps in fields ranging from architecture and engineering to education and manufacturing—proving that math turns everyday objects into precise knowledge."]
