Question: The volume of a cone is \(48\pi\) cubic units, and its height is \(9\) units. Find the radius of the base of the cone.
["Finding the Radius of a Cone When Volume and Height Are Known", "Understanding how to calculate the dimensions of geometric shapes is essential in mathematics, engineering, and everyday problem-solving. One common scenario involves finding the radius of a cone’s base when the cone’s volume and height are known. In this article, we’ll explore a typical problem where the volume of a cone is (48\pi) cubic units and its height is (9) units — and we’ll determine the radius of the base step by step.", "---", "### The Volume Formula for 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) = volume\n- (r) = radius of the base\n- (h) = height of the cone", "This formula reflects that a cone occupies one-third the volume of a cylinder with the same base and height.", "---", "### Step-by-Step Solution", "We are given:\n- (V = 48\pi)\n- (h = 9)", "Substitute these values into the volume formula:", "[\n48\pi = \frac{1}{3} \pi r^2 (9)\n]", "Simplify the right-hand side:", "[\n48\pi = 3\pi r^2\n]", "Now, divide both sides by (\pi) to eliminate (\pi):", "[\n48 = 3r^2\n]", "Next, divide both sides by 3:", "[\nr^2 = \frac{48}{3} = 16\n]", "Finally, take the square root:", "[\nr = \sqrt{16} = 4\n]", "Since radius cannot be negative, we discard the negative solution.", "---", "### Conclusion", "The radius of the base of the cone is 4 units. This straightforward calculation demonstrates how geometric formulas empower us to easily recover missing dimensions, making it a fundamental skill in STEM fields and practical applications.", "If you often face questions involving cone volumes, remember the key formula: (V = \frac{1}{3} \pi r^2 h), and practice solving by substituting known values and isolating (r).", "---", "Keywords for SEO:\n- Cone volume formula\n- Find radius of cone given volume and height\n- How to calculate radius from cone volume\n- Cone geometry problems\n- Geometry problem solving\n- Volume of a cone calculation", "Optimized for search engines with targeted keywords, this article explains clearly and concisely how to solve cone radius problems using real numbers and step-by-step logic."]
