When the radius is doubled to 10 cm, the new volume \( V_2 \) is:
["# When the Radius Is Doubled to 10 cm, the New Volume ( V_2 ) Is", "When working with geometric shapes involving cylindrical or spherical volumes, the radius plays a crucial role in determining the final volume. A common question in geometry is: What happens to the volume when the radius is doubled? Let’s explore this clearly, focusing on a cylinder as a practical example—and specifically, what happens when the radius increases to 10 cm, revealing the new volume ( V_2 ).", "## Understanding Volume in a Cylinder", "The volume ( V ) of a cylinder is calculated using the formula:", "[\nV = \pi r^2 h\n]", "Where:\n- ( r ) = radius of the base\n- ( h ) = height of the cylinder", "Notice that volume depends on the square of the radius. This means doubling the radius does not simply double the volume—it multiplies the volume by four.", "## Doubling the Radius to 10 cm", "Suppose we start with a cylinder where the original radius ( r ) is not 10 cm but is halved to allow straightforward doubling later. Let’s assume that after scaling, the radius becomes ( r_2 = 10 ) cm.", "Using the volume formula for clarity:\n- Original radius: ( r_1 )\n- After doubling: ( r_2 = 10 ) cm = ( 2r_1 ) → So, ( r_1 = 5 ) cm", "Now compute the volume before scaling:\n[\nV_1 = \pi (5)^2 h = \pi \ imes 25 \ imes h = 25\pi h\n]", "Now compute the new volume ( V_2 ) with radius 10 cm:\n[\nV_2 = \pi (10)^2 h = \pi \ imes 100 \ imes h = 100\pi h\n]", "## Compare Volumes", "Compare ( V_1 ) and ( V_2 ):\n[\n\frac{V_2}{V_1} = \frac{100\pi h}{25\pi h} = 4\n]", "So, doubling the radius increases the volume by a factor of 4. That is:\n[\nV_2 = 4 \ imes V_1\n]", "## What This Means in Real Applications", "- Doubling the radius (to 10 cm) quadruples the volume, not doubles it.\n- This principle applies to spheres, cylinders, and other volume-formula shapes where volume depends on ( r^2 ).\n- Understanding this helps in scaling models, containers, bottles, fuel tanks, and storage systems effectively.", "## Conclusion", "If the radius is doubled to 10 cm, the new volume ( V_2 ) is four times the original volume ( V_1 ). This reflects the squaring relationship in the volume formula: ( V \propto r^2 ). Recognizing this pattern ensures accurate estimations when scaling geometric forms, especially in engineering, design, and science.", "---", "Key Takeaways:\n- Radius influences volume quadratically.\n- Doubling radius → volume becomes 4× larger.\n- Useful for volume calculations in real-world applications.", "If you're working with similar problems, remember: radius squared determines volume—double the radius → quadruple the volume!"]
