Question: A science fair judge is evaluating a student's model of a planet, shaped like a sphere with radius $ 2r $, and a supporting half-sphere with radius $ r $. What is the ratio of the volume of the planet to the volume of the half-sphere?
["Title: Mastering Volume Ratios: A Science Fair Project on Spheres and Half-Spheres", "Meta Description:\nLearn how to calculate the ratio of the volume of a full sphere (planet) with radius $ 2r $ to a supporting hemisphere (half-sphere) with radius $ r $. This science fair insight explains the key volume formulas and derived ratio.", "---", "### Introduction", "Science fair projects offer exciting opportunities to explore fundamental mathematical and physical concepts. One fascinating project involves a student model combining a spherical planet — with radius $ 2r $ — and a supporting hemispherical base — shaped as half of a sphere with radius $ r $. A key evaluation question is: What is the ratio of the volume of the planet to the volume of the supporting half-sphere?", "This article breaks down the problem step-by-step, applying core volume formulas and revealing the elegant relationship between these geometric shapes.", "---", "### Understanding the Shapes", "1. Sphere (Planet):\n A full sphere of radius $ 2r $ has volume given by the standard formula:\n [\n V_{\ ext{sphere}} = \frac{4}{3}\pi (2r)^3\n ]\n Calculating this:\n [\n V_{\ ext{sphere}} = \frac{4}{3}\pi (8r^3) = \frac{32}{3}\pi r^3\n ]", "2. Hemisphere (Half-Sphere Base):\n This curved flat base is exactly half of a sphere with radius $ r $, so its volume is half the volume of a full sphere of radius $ r $:\n [\n V_{\ ext{hemisphere}} = \frac{1}{2} \ imes \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3\n ]", "---", "### Finding the Volume Ratio", "We are asked to compute the ratio of the planet’s volume to the supporting hemisphere’s volume:\n[\n\ ext{Ratio} = \frac{V_{\ ext{sphere}}}{V_{\ ext{hemisphere}}} = \frac{\frac{32}{3}\pi r^3}{\frac{2}{3}\pi r^3}\n]", "Simplify by canceling common terms ($\frac{\pi r^3}{3}$):\n[\n\ ext{Ratio} = \frac{32}{2} = 16\n]", "---", "### Conclusion", "The volume of the planet — a sphere of radius $ 2r $ — is 16 times greater than the volume of the supporting half-sphere with radius $ r $. This ratio highlights how slight changes in size dramatically affect three-dimensional space.", "For science fair judges, this project clearly demonstrates:", "- Accurate application of geometric volume formulas\n- Understanding of partial volumes like hemispheres\n- Clear communication of proportional reasoning through real-world modeling", "This is a standout example of how math brings planetary science and engineering models to life.", "---", "Keywords:\nscience fair project, volume ratio, sphere volume, hemisphere volume, geometry, planet model, science fair evaluation, radius $ 2r $, radius $ r $, math modeling, teacher resources, student project inspiration"]
