To find the volume of Kairay's sculpture, we need to calculate the volume of the larger hemisphere and subtract the volume of the smaller hemisphere.
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
- Volume of the larger hemisphere (radius = 20 inches):
\[ V_{\text{large}} = \frac{2}{3} \times 3.14 \times (20)^3 \]
Calculating \( (20)^3 \):
\[ (20)^3 = 8000 \]
Now, substituting it into the volume formula:
\[ V_{\text{large}} = \frac{2}{3} \times 3.14 \times 8000 \]
Calculating \( \frac{2}{3} \times 3.14 \):
\[ \frac{2}{3} \times 3.14 \approx 2.0933 \]
Now multiply by 8000:
\[ V_{\text{large}} \approx 2.0933 \times 8000 \approx 16746.4 \]
So the volume of the larger hemisphere is approximately 16746.4 cubic inches.
- Volume of the smaller hemisphere (radius = 15 inches):
\[ V_{\text{small}} = \frac{2}{3} \times 3.14 \times (15)^3 \]
Calculating \( (15)^3 \):
\[ (15)^3 = 3375 \]
Substituting into the volume formula:
\[ V_{\text{small}} = \frac{2}{3} \times 3.14 \times 3375 \]
Calculating \( \frac{2}{3} \times 3.14 \):
\[ \frac{2}{3} \times 3.14 \approx 2.0933 \]
Now multiply by 3375:
\[ V_{\text{small}} \approx 2.0933 \times 3375 \approx 7070.625 \]
So the volume of the smaller hemisphere is approximately 7070.625 cubic inches.
- Total volume of the sculpture:
Now we subtract the volume of the smaller hemisphere from the volume of the larger hemisphere:
\[ V_{\text{s sculpture}} = V_{\text{large}} - V_{\text{small}} \]
\[ V_{\text{s sculpture}} \approx 16746.4 - 7070.625 \approx 9675.775 \]
Rounding to the nearest whole number:
\[ V_{\text{s sculpture}} \approx 9676 \]
So the volume of Kairay's sculpture is approximately \( 9,676 \) cubic inches, which is closest to the provided option:
9,682 in.³.