A cylinder has a radius of 3 cm and a height of 10 cm. If the height is increased by 50% and the radius is decreased by 20%, what is the new volume of the cylinder?
["Understanding Cylinder Volume: Calculating the New Volume After Modifications", "The volume of a cylinder is a fundamental concept in geometry, widely applied in engineering, architecture, and everyday science. The classic formula for the volume ( V ) of a cylinder is:", "[\nV = \pi r^2 h\n]", "where ( r ) is the radius and ( h ) is the height. In this article, we’ll explore how changes in these dimensions affect the volume—using a real-world example to illustrate key mathematical principles.", "---", "### The Original Cylinder’s Dimensions", "We start with a cylinder with:", "- Radius ( r = 3 ) cm\n- Height ( h = 10 ) cm", "First, compute the original volume:", "[\nV_{\ ext{original}} = \pi (3)^2 (10) = \pi \ imes 9 \ imes 10 = 90\pi \ ext{ cm}^3\n]", "---", "### Adjusting the Dimensions", "Now, apply the specified changes:", "1. Increase the height by 50%:\n A 50% increase means multiplying by 1.5:\n [\n h_{\ ext{new}} = 10 \ imes 1.5 = 15 \ ext{ cm}\n ]", "2. Decrease the radius by 20%:\n A 20% reduction means the remaining radius is 80% of the original:\n [\n r_{\ ext{new}} = 3 \ imes (1 - 0.20) = 3 \ imes 0.8 = 2.4 \ ext{ cm}\n ]", "---", "### Calculating the New Volume", "Substitute the revised radius and height into the volume formula:", "[\nV_{\ ext{new}} = \pi (2.4)^2 (15)\n]", "First compute ( (2.4)^2 = 5.76 ), then:", "[\nV_{\ ext{new}} = \pi \ imes 5.76 \ imes 15 = \pi \ imes 86.4 \approx 86.4\pi \ ext{ cm}^3\n]", "---", "### The Final Result", "The new volume after modifying the dimensions is:", "[\n\boxed{86.4\pi \ ext{ cm}^3}\n]", "This result shows how proportional changes in a cylinder’s dimensions—such as increasing height while reducing radius—directly affect the stored volume, making it crucial to apply relative scaling correctly.", "---", "Knowing how to calculate volume changes helps in practical scenarios like designing containers, optimizing fluid storage, and understanding physical properties in materials science. Whether you're a student, engineer, or DIY enthusiast, mastering these calculations supports better decision-making with tangible objects.", "If you're interested in further math tips and geometry insights, explore more STEM resources to strengthen your numerical skills!"]
