Question Nabhitha measure the volume of a sink basin by modeling it as a hemisphere. Nabhitha measures its radius to be \( 15 \frac{1}{4} \) inches. Find the sink's volume in cubic inches. Round your answer to the nearest tenth if necessary.

To calculate the volume of the sink basin modeled as a hemisphere, we can use the formula for the volume of a hemisphere, which is given by \( V = \frac{2}{3} \pi r^3 \). First, we need to convert the mixed number radius into an improper fraction: \( r = 15 \frac{1}{4} = \frac{61}{4} \) inches. Now we can substitute this value into the volume formula: \[ V = \frac{2}{3} \pi \left(\frac{61}{4}\right)^3. \] Calculating \( \left(\frac{61}{4}\right)^3 = \frac{226981}{64} \). Substituting this back into the volume formula gives: \[ V = \frac{2}{3} \pi \left(\frac{226981}{64}\right) \approx \frac{2 \times 226981 \times \pi}{192} \approx \frac{453962 \pi}{192} \approx 2366.38 \text{ cubic inches}. \] Rounding to the nearest tenth, the sink's volume is approximately \( 2366.4 \) cubic inches. To double-check these calculations, you could also visualize using cross-sectional geometry of a sphere. The hemisphere represents half of the entire sphere, so imagining how deep it would hold water could offer insights into its rounded edges and capacity. It’s a fun way to connect mathematics to everyday objects! If you're fascinated by how volume calculations relate to interior design, remember that many sink basin shapes are inspired by geometric design principles! Understanding the relationship between cubic measurements and practical applications can assist in selecting kitchen or bath sinks that fit both aesthetics and functionality.