#### 42.5Question: The height of a cone is tripled and its radius is halved. What is the ratio of the new volume to the original volume?
["# The Height of a Cone Is Tripled—And Its Radius Halved: What’s the Volume Ratio?", "In a mobile-first world where clever math and geometry matter more than ever, a simple yet intriguing question is circulating in smart circles: What happens to a cone’s volume when its height triples and its radius shrinks to half its original size? For those navigating technical frontiers or just curious about how shapes transform, this ratio offers valuable insight beyond the surface. Understanding this shift helps inform design decisions, cost modeling, or even intuitive predictions—especially where precision matters in fields like manufacturing, architecture, or educational content. As digital tools emphasize data literacy, mastering this concept keeps users ahead, whether exploring niche trends or optimizing real-world calculations.", "## Why This Question Is Rising in US Conversations", "Outside academic circles, this cone-related ratio is quietly gaining traction across US online communities focused on STEM, design, and data-driven decision-making. With growing interest in spatial reasoning and scalable systems, the contrast between tripling height and halving radius offers a relatable problem many can visualize through everyday examples—from kitchen containers to industrial prototypes. People often discuss these changes while problem-solving on mobile devices, seeking clear answers that avoid surprises. The simplicity of geometry makes it ideal for Discover searches centered on learning, curiosity, and practical understanding—especially amid rising demand for intuitive, trustworthy info in a fast-paced digital environment.", "## How to Actually Calculate the New Volume Ratio", "To uncover the ratio of new volume to original volume, start with the formula for a cone’s volume: \n\[ V = \frac{1}{3} \pi r^2 h \]", "For the original cone: \n\[ V_{\ ext{original}} = \frac{1}{3} \pi r^2 h \]", "Now apply the changes: height becomes \( 3h \), radius becomes \( \frac{r}{2} \): \n\[ V_{\ ext{new}} = \frac{1}{3} \pi \left( \frac{r}{2} \right)^2 (3h) \]", "Simplify the expression: \n\[ V_{\ ext{new}} = \frac{1}{3} \pi \left( \frac{r^2}{4} \right) (3h) = \frac{1}{3} \pi r^2 h \cdot \frac{3}{4} \]", "Divide new volume by original: \n\[ \frac{V_{\ ext{new}}}{V_{\ ext{original}}} = \frac{\frac{1}{3} \pi r^2 h \cdot \frac{"]
