A cylinder has a height twice its radius. If its volume is 288π cm³, find the radius.
["Title: How to Find the Radius of a Cylinder When Volume and Height Are Related – Solving a Real-World Problem with Cylinder Geometry", "Cylinders are essential geometric shapes found in everyday life—from cans and pipes to water tanks and engineering components. Understanding how to calculate the radius, especially when height relates directly to radius, is crucial for students, architects, and DIY enthusiasts. In this article, we’ll explore a specific problem: a cylinder whose height is twice its radius and has a volume of 288π cm³. We’ll walk through the steps to find the radius, using solid geometry combined with algebra.", "### Understanding the Cylinder Volume Formula", "The volume ( V ) of a right circular cylinder is given by:", "[\nV = \pi r^2 h\n]", "where:\n- ( r ) = radius of the base (in cm),\n- ( h ) = height (in cm).", "In this problem, we are told the cylinder’s height is twice the radius, so:", "[\nh = 2r\n]", "The volume is given as:", "[\nV = 288\pi \ ext{ cm}^3\n]", "### Substituted into the Formula", "Start by substituting ( h = 2r ) into the volume formula:", "[\nV = \pi r^2 (2r) = 2\pi r^3\n]", "Set this equal to the known volume:", "[\n2\pi r^3 = 288\pi\n]", "### Solving for ( r )", "First, divide both sides by ( \pi ) to eliminate the constant:", "[\n2r^3 = 288\n]", "Next, divide both sides by 2:", "[\nr^3 = 144\n]", "Now, take the cube root of both sides:", "[\nr = \sqrt[3]{144}\n]", "While 144 isn’t a perfect cube, we can simplify or approximate:", "[\n144 = 12 \ imes 12 = 2^4 \cdot 3^2\n]", "But for practical purposes, use a calculator:", "[\nr \approx 5.24 \ ext{ cm (approximately)}\n]", "However, if an exact form is preferred:", "[\nr = \sqrt[3]{144}\n]", "But since the problem involves integer-like volume and integer relationship, check: could ( r = 6 )?", "Try ( r = 6 ):", "[\nh = 2 \cdot 6 = 12\n]", "[\nV = \pi (6)^2 (12) = \pi \cdot 36 \cdot 12 = 432\pi \quad \ ext{(too high)}\n]", "Try ( r = 4 ):", "[\nh = 8\n]", "[\nV = \pi \cdot 16 \cdot 8 = 128\pi \quad \ ext{(too low)}\n]", "So, the exact solution is:", "[\nr = \sqrt[3]{144} \ ext{ cm}\n]", "But since ( 144 = 8 \cdot 18 ),\n[\nr = \sqrt[3]{8 \cdot 18} = 2\sqrt[3]{18}\n]", "Though simplified, the clean exact value is best left as:", "[\n\boxed{r = \sqrt[3]{144} \ ext{ cm}}\n]", "For practical use, approximately:", "[\n\boxed{r \approx 5.24 \ ext{ cm}}\n]", "### Why This Problem Matters", "Knowing how to solve such problems bridges geometry and algebra — essential in fields like construction, manufacturing, and interior design. When height = 2 × radius, the relationship simplifies volume calculations, saving time and minimizing errors in real-world applications. Whether designing a storage tank or selecting materials, mastering these principles ensures accurate and efficient planning.", "---", "Keywords: cylinder volume formula, radius of cylinder, cylinder height twice radius, solve cylinder radius, volume of cylinder with height formula, solid geometry problems, algebra and geometry, cylinder calculation tips", "Meta Description:\nLearn how to find the radius of a cylinder when height is twice the radius and volume is 288π cm³. Step-by-step solution with exact formula and practical application. Ideal for students and DIY builders.", "Anchors: cylinder volume problems, how to calculate cylinder radius, cylinder geometry tutorial, solve geometry problems with algebra"]
