To calculate the volume of Kairay's sculpture, we need to find the volume of the larger hemisphere and then subtract the volume of the smaller hemisphere from it.
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Step 1: Calculate the volume of the larger hemisphere (radius = 20 inches)
Using \( r = 20 \) inches:
\[ V_{large} = \frac{2}{3} \cdot 3.14 \cdot (20)^3 \]
Calculating \( (20)^3 \):
\[ 20^3 = 8000 \]
Now substituting back into the volume formula:
\[ V_{large} = \frac{2}{3} \cdot 3.14 \cdot 8000 \] \[ V_{large} = \frac{2 \cdot 3.14 \cdot 8000}{3} \] \[ V_{large} = \frac{50240}{3} \approx 16746.67 \]
Step 2: Calculate the volume of the smaller hemisphere (radius = 15 inches)
Using \( r = 15 \) inches:
\[ V_{small} = \frac{2}{3} \cdot 3.14 \cdot (15)^3 \]
Calculating \( (15)^3 \):
\[ 15^3 = 3375 \]
Now substituting back into the volume formula:
\[ V_{small} = \frac{2}{3} \cdot 3.14 \cdot 3375 \] \[ V_{small} = \frac{2 \cdot 3.14 \cdot 3375}{3} \] \[ V_{small} = \frac{21225}{3} \approx 7075 \]
Step 3: Calculate the volume of the sculpture
Now, we subtract the volume of the smaller hemisphere from the volume of the larger hemisphere:
\[ V_{sculpture} = V_{large} - V_{small} \] \[ V_{sculpture} \approx 16746.67 - 7075 \approx 9671.67 \]
Rounding to the nearest whole number, we find:
\[ V_{sculpture} \approx 9682 \text{ in}^3 \]
Thus, the volume of Kairay's sculpture is approximately 9,682 inches cubed.
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