Question: A spherical buoy used by an oceanographer has a radius of $ 2x $, while a hemisphere-shaped sensor has a radius of $ 3x $. What is the ratio of their volumes?
["Understanding the Volume Ratio: A Spherical Buoy vs. a Hemisphere Sensor", "When studying ocean ecosystems, precise measurement tools are essential—and understanding their relative volumes helps oceanographers design better experiments and equipment. A common question in marine research involves comparing the volumes of two key devices: a spherical buoy used for long-term buoyancy and data transmission, and a hemisphere-shaped sensor used in underwater monitoring.", "### The Geometry at Play", "Let’s begin by recalling the formulas for volume:", "- Volume of a sphere with radius $ r $ is:\n [\n V_{\ ext{sphere}} = \frac{4}{3} \pi r^3\n ]", "- Volume of a hemisphere is half that of a full sphere:\n [\n V_{\ ext{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3\n ]", "Now analyze the two devices:", "1. Spherical Buoy\n Radius = $ 2x $\n Volume:\n [\n V_{\ ext{buoy}} = \frac{4}{3} \pi (2x)^3 = \frac{4}{3} \pi (8x^3) = \frac{32}{3} \pi x^3\n ]", "2. Hemisphere Sensor\n Radius = $ 3x $\n Volume:\n [\n V_{\ ext{sensor}} = \frac{2}{3} \pi (3x)^3 = \frac{2}{3} \pi (27x^3) = \frac{54}{3} \pi x^3 = 18 \pi x^3\n ]", "### Calculating the Volume Ratio", "To find the ratio of the buoy’s volume to the sensor’s volume:", "[\n\ ext{Ratio} = \frac{V_{\ ext{buoy}}}{V_{\ ext{sensor}}} = \frac{\frac{32}{3} \pi x^3}{18 \pi x^3}\n]", "Cancel $ \pi x^3 $ from numerator and denominator:\n[\n= \frac{32}{3} \div 18 = \frac{32}{3} \cdot \frac{1}{18} = \frac{32}{54} = \frac{16}{27}\n]", "### Final Answer", "The ratio of the volume of the spherical buoy (radius $ 2x $) to the hemisphere-shaped sensor (radius $ 3x $) is:\n[\n\boxed{\frac{16}{27}}\n]", "This means the buoy’s volume is smaller relative to the sensor’s, which is crucial when planning buoyancy, floatation duration, and sensor deployment in marine research. Understanding such volume ratios ensures efficient design and accurate data collection in oceanography."]
