Question: A spherical data storage unit has a radius of $ y $ units, and a hemispherical backup dome has a radius of $ 3y $ units. What is the ratio of the storage unit's volume to the backup dome's volume?
["Title: Comparing Storage Volumes: A Spherical Data Unit vs. a Hemispherical Backup Dome", "When designing efficient data storage systems, understanding the volume capacity of storage components is essential—especially when distinguishing between primary storage units and backup enclosures. In this analysis, we explore the volume ratio between a spherical data storage unit with radius $ y $ units and a hemispherical backup dome with radius $ 3y $ units.", "### Spherical Data Storage Unit", "A sphere’s volume is calculated using the formula:\n$$\nV_{\ ext{sphere}} = \frac{4}{3}\pi r^3\n$$\nFor the data storage unit with radius $ y $:\n$$\nV_{\ ext{storage}} = \frac{4}{3}\pi y^3\n$$", "### Hemispherical Backup Dome", "A hemisphere is half the volume of a full sphere. The volume of a full sphere with radius $ 3y $ is:\n$$\nV_{\ ext{full sphere}} = \frac{4}{3}\pi (3y)^3 = \frac{4}{3}\pi (27y^3) = 36\pi y^3\n$$\nThus, the volume of the hemispherical dome is:\n$$\nV_{\ ext{backup}} = \frac{1}{2} \ imes 36\pi y^3 = 18\pi y^3\n$$", "### Volume Ratio", "Now, compute the ratio of the spherical storage volume to the hemispherical backup volume:\n$$\n\ ext{Ratio} = \frac{V_{\ ext{storage}}}{V_{\ ext{backup}}} = \frac{\frac{4}{3}\pi y^3}{18\pi y^3}\n$$\nCancel $ \pi y^3 $ from numerator and denominator:\n$$\n= \frac{\frac{4}{3}}{18} = \frac{4}{3} \ imes \frac{1}{18} = \frac{4}{54} = \frac{2}{27}\n$$", "### Conclusion", "The ratio of the volume of the spherical data storage unit (radius $ y $) to the hemispherical backup dome (radius $ 3y $) is exactly:\n$$\n\boxed{\frac{2}{27}}\n$$\nThis means the spherical unit holds significantly less volume—about 1/27th—despite its relatively modest radius increase compared to the much larger backup dome. Understanding such volume scaling helps engineers optimize space, cooling, and energy use in data centers."]
