A cone has a volume of \( 100\pi \) cubic centimeters and a height of 12 cm. What is the radius of the base?
["# Find the Radius of a Cone with Volume ( 100\pi ) cm³ and Height 12 cm", "Understanding the volume of a cone is essential for solving problems in geometry, engineering, architecture, and science. If you’ve come across a cone with a known volume and height, finding the radius of its base is a natural calculation to perform. In this article, we’ll walk through the process step-by-step using the formula for the volume of a cone, apply it to a real scenario, and determine the exact radius of the base.", "---", "## The Formula for the Volume of a Cone", "The volume ( V ) of a cone is given by the formula:", "[\nV = \frac{1}{3} \pi r^2 h\n]", "Where:\n- ( V ) is the volume,\n- ( r ) is the radius of the base,\n- ( h ) is the height of the cone,\n- ( \pi ) is a mathematical constant (~3.14159).", "---", "## Given Values", "From the problem, we know:\n- Volume ( V = 100\pi ) cm³,\n- Height ( h = 12 ) cm.", "We are tasked with finding the radius ( r ).", "---", "## Step-by-Step Calculation", "### Step 1: Substitute known values into the volume formula", "[\n100\pi = \frac{1}{3} \pi r^2 (12)\n]", "### Step 2: Simplify the equation", "First, multiply ( \frac{1}{3} \ imes 12 = 4 ), so:", "[\n100\pi = 4\pi r^2\n]", "### Step 3: Eliminate ( \pi ) from both sides", "Since ( \pi ) appears on both sides, divide both sides by ( \pi ):", "[\n100 = 4r^2\n]", "### Step 4: Solve for ( r^2 )", "Divide both sides by 4:", "[\nr^2 = \frac{100}{4} = 25\n]", "### Step 5: Take the square root", "[\nr = \sqrt{25} = 5\n]", "---", "## Final Answer", "The radius of the base of the cone is ( 5 ) centimeters.", "---", "## Why This Matters", "Knowing the radius from the volume and height helps in real-world applications such as:\n- Calculating the capacity of conical containers,\n- Estimating material needs in manufacturing,\n- Solving geometric and engineering design problems.", "Using the volume formula directly ensures accurate results without unnecessary complications—especially when height and volume are already provided.", "---", "## Summary", "Given a cone with volume ( 100\pi ) cm³ and height 12 cm, solving for the radius yields ( r = 5 ) cm. This straightforward calculation using ( V = \frac{1}{3}\pi r^2 h ) highlights how geometry underpins both theoretical and practical problem-solving.", "---", "Keywords: Volume of a cone, formula cone volume, find radius cone, ( V = \frac{1}{3}\pi r^2 h ), solve geometry problem, radius of cone, center radius calculation."]
