A cone has a volume of 150 cubic centimeters and a height of 9 centimeters. What is the radius of its base?
["Finding the Radius of a Cone’s Base: A Step-by-Step Guide with Volume 150 cm³ and Height 9 cm", "Understanding the volume of a cone is essential in geometry, architecture, engineering, and everyday applications. Whether you’re designing a cone-shaped container or solving math problems, knowing how to calculate the radius from volume and height is a vital skill. In this article, we’ll explore how to find the base radius of a cone given its volume and height—using a real-world example with a volume of 150 cubic centimeters and a height of 9 centimeters.", "### What is the Volume of a Cone?", "The formula for the volume ( V ) of a cone is:", "[\nV = \frac{1}{3} \pi r^2 h\n]", "where:\n- ( r ) is the radius of the base\n- ( h ) is the height\n- ( \pi \approx 3.1416 ) (a mathematical constant)", "This formula shows that volume depends directly on the area of the circular base (( \pi r^2 )) and the cone’s height.", "### Step 1: Plug in Known Values", "We are given:\n- Volume ( V = 150 , \ ext{cm}^3 )\n- Height ( h = 9 , \ ext{cm} )", "Substitute into the formula:", "[\n150 = \frac{1}{3} \pi r^2 (9)\n]", "### Step 2: Simplify the Equation", "First, simplify the right-hand side:", "[\n150 = 3\pi r^2\n]", "Now isolate ( r^2 ):", "[\nr^2 = \frac{150}{3\pi} = \frac{50}{\pi}\n]", "### Step 3: Calculate ( r^2 ) Using ( \pi \approx 3.1416 )", "[\nr^2 \approx \frac{50}{3.1416} \approx 15.915\n]", "### Step 4: Solve for ( r )", "Take the square root:", "[\nr \approx \sqrt{15.915} \approx 3.99 , \ ext{cm}\n]", "Rounded to two decimal places, the radius is approximately 4.00 cm.", "### Final Answer", "A cone with a volume of 150 cm³ and a height of 9 cm has a base radius of approximately 4.00 centimeters.", "This calculation is essential not only in theoretical mathematics but also in practical engineering, manufacturing, and design where precise cone geometry is required.", "---", "Keywords: cone volume formula, find radius of cone, geometry problem, cone base radius, volume calculation, 3D shapes, math tutorial, π cone volume, practical geometry", "Meta Description: Learn how to calculate the radius of a cone’s base using volume and height. Step-by-step guide with a 150 cm³ cone and 9 cm height. Result: radius ≈ 4.00 cm.", "Tags: #ConeGeometry #VolumeFormula #MathHelp #EngineeringBasics #3DShapes #RadiusCalculation"]
