Question: A science administrator is reviewing a grant proposal involving a cylindrical tank with radius $ 3r $ and height $ 4r $, and a cube with edge length $ 6r $. What is the ratio of the volume of the cylinder to the volume of the cube?
["Title: Understanding the Volume Ratio: Cylinder vs. Cube in Scientific Grant Proposals", "When evaluating scientific proposals involving geometric components—such as storage tanks, containment vessels, or material containers—understanding the volume ratios between different shapes is crucial. In one recent grant application, a reviewer was tasked with comparing the volume of a cylindrical tank to that of a cube, both constructed using a uniform scaling factor. This article explores the mathematical analysis behind the ratio of the volume of a cylinder with radius $ 3r $ and height $ 4r $ to that of a cube with edge length $ 6r $, offering insight into how such comparisons enhance clarity and feasibility in scientific planning.", "---", "### Step 1: Volume of the Cylinder", "The formula for the volume $ V $ of a cylinder is:", "[\nV_{\ ext{cylinder}} = \pi r_{\ ext{base}}^2 \cdot h\n]", "Given:\n- Radius $ r_{\ ext{base}} = 3r $\n- Height $ h = 4r $", "Substituting:", "[\nV_{\ ext{cylinder}} = \pi (3r)^2 \cdot 4r = \pi (9r^2) \cdot 4r = 36\pi r^3\n]", "---", "### Step 2: Volume of the Cube", "The formula for the volume of a cube is:", "[\nV_{\ ext{cube}} = s^3\n]", "Given edge length $ s = 6r $:", "[\nV_{\ ext{cube}} = (6r)^3 = 216r^3\n]", "---", "### Step 3: Compute the Volume Ratio", "Now, compute the ratio of the cylinder’s volume to the cube’s volume:", "[\n\ ext{Ratio} = \frac{V_{\ ext{cylinder}}}{V_{\ ext{cube}}} = \frac{36\pi r^3}{216r^3} = \frac{36\pi}{216} = \frac{\pi}{6}\n]", "Thus, the ratio is:", "[\n\frac{\pi}{6}\n]", "---", "### Why This Ratio Matters in Scientific Contexts", "Understanding such volume relationships allows scientists and engineers to assess spatial efficiency, resource allocation, and containment capacity. For instance, the cylindrical tank holds $ 36\pi r^3 $, while the cube offers $ 216r^3 $. Though the cube has a larger absolute volume, the cylinder’s compact design and favorable height-to-radius ratio may offer practical advantages in construction, material use, or spatial integration—factors critical to grant-funded projects involving infrastructure or experimental setups.", "Moreover, dimensional consistency—here using the same variable $ r $—ensures proportionality and simplifies comparative analysis, a hallmark of rigorously planned scientific proposals.", "---", "### Conclusion", "The ratio of the volume of the cylinder (radius $ 3r $, height $ 4r $) to the volume of the cube (edge $ 6r $) is:", "[\n\boxed{\frac{\pi}{6}}\n]", "This insight demonstrates how fundamental geometric principles support effective scientific evaluation, ensuring that funding supports both theoretical soundness and practical feasibility."]
