Question: A cone has a base radius of $ 2a $ and height $ 3a $. A smaller cone is cut off parallel to the base, leaving a frustum with height $ 2a $. What is the ratio of the volume of the small cone to the original cone?
["Why Curious Minds Are Exploring the Geometry of Cone Frustums", "Have you ever wondered why certain shapes shrink so predictably when part cut away? In everyday design, engineering, and even consumer products, understanding how trimming cones affects volume reveals fascinating math behind seemingly simple forms. Right now, a clear, curious audience is studying conical geometry—especially when frustums and proportional removal create precise volume ratios. This topic stirs interest not only among STEM learners but also professionals navigating spatial design, packaging, or structural efficiency. What’s the real story behind shrinking a cone parallel to its base, leaving a defined frustum? The answer lies in volume ratios that reflect fundamental mathematical consistency—offering surprising clarity in a complex-looking problem.", "---", "A Question Resonating in Design and Engineering Circles", "The question at the heart of this exploration is clear: A cone has base radius $2a$ and height $3a$. A smaller cone is removed parallel to the base, leaving a frustum with total height $2a$. What is the ratio of the small cone’s volume to the original? This query gains traction amid growing digital discussions around spatial efficiency, material use, and scalable design. As industries refine precision in manufacturing and 3D modeling, such geometric proportions become vital for optimizing space and resources—making the problem both relevant and timeless.", "---", "How the Small Cone Relates to the Original Volume", "To understand the ratio, begin by recalling the formula for a cone’s volume: \n$$ V = \frac{1}{3} \pi r^2 h $$", "For the original cone: \n- Base radius $ r = 2a $ \n- Height $ h = 3a $ \nSo, \n$$ V_{\ ext{original}} = \frac{1}{3} \pi (2a)^2 (3a) = \frac{1}{3} \pi (4a^2)(3a) = 4\pi a^3 $$", "When"]
