Question: A linguist models language divergence over time using a spherical model of evolving dialects, where each dialect occupies a spherical volume with radius $2y$. A related simplified model uses a hemisphere with radius $3y$ to represent a branching phenomenon. What is the ratio of the volume of the hemisphere to the volume of the sphere? Express your answer as a simplified fraction.

["Understanding Language Evolution Through Volume: A Simplified Approach", "Ever wondered how languages grow and split over time? Imagine each dialect as a growing space shaped by history and geography—linguists often use models to capture this evolution visually. One such model shrinks dialects into distinct spheres, each with radius $2y$, to represent isolated linguistic communities. But what if we tried a different shape—like a hemisphere—to reflect a branching branching process? This simplified spherical model uses a hemisphere with radius $3y$ to symbolize how language division can spread beyond a single point. What does math reveal when comparing this hemisphere’s volume to a full sphere? And why does this matter for science, technology, or cultural trends?", "Why This Concept Is Gaining Attention in the US", "Understanding how language evolves isn’t just academic—it’s increasingly relevant in a rapidly globalizing world. With rising interest in AI, cultural preservation, and digital communication patterns, models like the spherical dialect map offer intuitive ways to visualize complex social dynamics. The idea of representing evolving dialects through geometric volumes appeals to both researchers and tech-savvy audiences navigating digital content. While not widely known, such models reflect growing curiosity about how language shapes identity and connectivity—especially amid demographic shifts and multicultural exchange across US communities.", "How This Ratio Credits Language and Math", "The question focuses on comparing two key volumes: \n- A full sphere with radius $2y$ representing a single dialect \n- A hemisphere with radius $3y$ symbolizing a branching dialect phenomenon", "The volume $V_{\ ext{sphere}}$ of a sphere is $\frac{4}{3}\pi r^3$. Substituting $r = 2y$: \n$$\nV_{\ ext{sphere}} = \frac{4}{3}\pi (2y)^3 = \frac{4}{3}\pi \cdot 8y^3 = \frac{32}{3}\pi y^3\n$$", "A hemisphere’s volume is half that of a full sphere: \n$$\nV_{\ ext{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3\n$$ \nWith $r = 3y$: \n$$\nV_{\ ext{hemisphere}} = \frac{2}{3}\pi (3y)^3 = \frac{2}{3}\pi \cdot 27y^3 = 18\pi y^3\n$$", "Now calculate the ratio of hemisphere volume to sphere volume: \n$$\n\frac{V_{\ ext{hemisphere}}}{V_{\ ext{sphere}}} = \frac{18\pi y"]