To calculate the volume of the sculpture formed by subtracting a smaller hemisphere from a larger hemisphere, we can use the formula for the volume of a hemisphere:
\[ V = \frac{2}{3} \pi r^3 \]
Step 1: Calculate the volume of the larger hemisphere (radius: 20 inches)
Using \( r = 20 \):
\[ V_{\text{large}} = \frac{2}{3} \pi (20)^3 \] \[ = \frac{2}{3} \pi (8000) \] \[ = \frac{16000}{3} \pi \]
Plugging in \( \pi \approx 3.14 \):
\[ V_{\text{large}} = \frac{16000}{3} \times 3.14 \]
Calculating \( \frac{16000 \times 3.14}{3} \):
\[ = \frac{50240}{3} \approx 16746.67 \text{ (approximately)} \]
Step 2: Calculate the volume of the smaller hemisphere (radius: 15 inches)
Using \( r = 15 \):
\[ V_{\text{small}} = \frac{2}{3} \pi (15)^3 \] \[ = \frac{2}{3} \pi (3375) \] \[ = \frac{6750}{3} \pi \]
Again, plugging in \( \pi \approx 3.14 \):
\[ V_{\text{small}} = \frac{6750}{3} \times 3.14 \]
Calculating \( \frac{6750 \times 3.14}{3} \):
\[ = \frac{21195}{3} \approx 7065 \text{ (approximately)} \]
Step 3: Calculate the volume of the sculpture
The volume of the sculpture is:
\[ V_{\text{sculpture}} = V_{\text{large}} - V_{\text{small}} \]
Using the approximations calculated:
\[ V_{\text{sculpture}} \approx 16746.67 - 7065 \approx 9681.67 \]
Final Calculation
Rounding to the nearest whole number:
\[ V_{\text{sculpture}} \approx 9682 \text{ cubic inches} \]
Thus, the volume of Kairay's sculpture is approximately 9,682 inches cubed.
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