To calculate the volume of Kairay's sculpture, we need to find the volume of the larger hemisphere and subtract the volume of the smaller hemisphere.
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.
- Calculate the volume of the larger hemisphere (radius = 20 inches):
\[ V_{large} = \frac{2}{3} \times 3.14 \times (20)^3 \]
Calculating \( (20)^3 \):
\[ 20^3 = 8000 \]
So,
\[ V_{large} = \frac{2}{3} \times 3.14 \times 8000 \]
Calculating this step-by-step:
\[ V_{large} = \frac{2 \times 3.14 \times 8000}{3} \]
\[ = \frac{50240}{3} \]
Calculating \( 50240 \div 3 \):
\[ V_{large} = 16746.67 , \text{in}^3 \]
- Calculate the volume of the smaller hemisphere (radius = 15 inches):
\[ V_{small} = \frac{2}{3} \times 3.14 \times (15)^3 \]
Calculating \( (15)^3 \):
\[ 15^3 = 3375 \]
So,
\[ V_{small} = \frac{2}{3} \times 3.14 \times 3375 \]
Calculating this step-by-step:
\[ V_{small} = \frac{2 \times 3.14 \times 3375}{3} \]
\[ = \frac{21165}{3} \]
Calculating \( 21165 \div 3 \):
\[ V_{small} = 7055 , \text{in}^3 \]
- Now, subtract the volume of the smaller hemisphere from the larger hemisphere:
\[ V_{sculpture} = V_{large} - V_{small} \]
Calculating:
\[ V_{sculpture} = 16746.67 - 7055 \]
\[ = 9691.67 , \text{in}^3 \]
Rounding to the nearest whole number gives:
\[ V_{sculpture} \approx 9692 , \text{in}^3 \]
From the options provided:
- 9,682 in.³ is the closest available answer.
Thus, the volume of Kairay's sculpture is approximately 9,682 inches cubed.