Question: The radius of a spherical pollen grain is $2y$ units, and a hemispherical spore has radius $3y$ units. What is the ratio of the volume of the pollen grain to the volume of the spore?
["Discover Hook: Why Do Scientists Care About Pollen and Spore Volumes? \nIn a world shaped by climate shifts, allergies, and natural ecosystems, understanding the physical properties of microscopic particles like pollen and spores is more relevant than ever—especially for researchers tracking environmental change and airborne allergens. A straightforward question about their volumes reveals surprising complexity. The radius of a spherical pollen grain is $2y$ units, and a hemispherical spore has radius $3y$. What does this ratio of volumes mean for science, health, and technology today? This isn’t just anatomy—it’s insight into nature’s engineering and its impact on human well-being.", "---", "Understanding the Volume Formula for Spheres and Hemispheres \nAt the core of calculating this ratio are standard geometric formulas: the volume of a full sphere is $\frac{4}{3}\pi r^3$, while a hemisphere is half that, or $\frac{2}{3}\pi r^3$. These formulas underpin countless applications—from agricultural modeling to medical diagnostics. When comparing volumes of different shapes, especially across biological particles, precision matters. Here, using $2y$ and $3y$ as radii creates a clear numerical comparison, revealing both scale and ratio without resorting to vague or controversial descriptions.", "---", "Calculating the Volume Ratio: Step by Step \nThe spherical pollen grain has radius $2y$. Its volume is: \n\[\nV_{\ ext{pollen}} = \frac{4}{3} \pi (2y)^3 = \frac{4}{3} \pi (8y^3) = \frac{32}{3} \pi y^3\n\]", "The hemispherical spore has radius $3y$. Its volume is half that of a full sphere: \n\[\nV_{\ ext{spore}} = \frac{1}{2} \cdot \frac{4}{3} \pi (3y)^3 = \frac{2}{3} \pi (27y^3) = \frac{54}{3} \pi y^3 = 18\pi y^3\n\]", "Therefore, the ratio of pollen to spore volume is: \n\[\n\ ext{Ratio} = \frac{V_{\ ext{pollen}}}{V_{\ ext{spore}}} = \frac{\frac{32}{3} \pi y^3}{18"]
