Kairay created a sculpture by forming a hemisphere with a radius of 20 inches and then removing a hemisphere with a radius of 15 inches from it. Calculate the volume of his sculpture. Use 3.14 as an approximation for pi. Round your answer to the nearest whole number.
To calculate the volume of the sculpture, we must find the volume of the larger hemisphere and subtract the volume of the smaller hemisphere that has been removed.
The volume of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.
First, let's calculate the volume of the larger hemisphere (with a radius of 20 inches):
\[ V_{\text{large}} = \frac{2}{3} \pi (20)^3 \]
\[ V_{\text{large}} = \frac{2}{3} \times 3.14 \times (20 \times 20 \times 20) \]
\[ V_{\text{large}} = \frac{2}{3} \times 3.14 \times 8000 \]
\[ V_{\text{large}} = \frac{2}{3} \times 25120 \]
\[ V_{\text{large}} = 16746.6667 \; \text{in}^3 \]
Next, we need to calculate the volume of the smaller hemisphere (with a radius of 15 inches):
\[ V_{\text{small}} = \frac{2}{3} \pi (15)^3 \]
\[ V_{\text{small}} = \frac{2}{3} \times 3.14 \times (15 \times 15 \times 15) \]
\[ V_{\text{small}} = \frac{2}{3} \times 3.14 \times 3375 \]
\[ V_{\text{small}} = \frac{2}{3} \times 10593.75 \]
\[ V_{\text{small}} = 7058.5 \; \text{in}^3 \]
Now, we subtract the volume of the smaller hemisphere from the volume of the larger hemisphere to obtain the volume of Kairay's sculpture:
\[ V_{\text{sculpture}} = V_{\text{large}} - V_{\text{small}} \]
\[ V_{\text{sculpture}} = 16746.6667 \; \text{in}^3 - 7058.5 \; \text{in}^3 \]
\[ V_{\text{sculpture}} = 9688.1667 \; \text{in}^3 \]
Rounded to the nearest whole number:
\[ V_{\text{sculpture}} \approx 9688 \; \text{in}^3 \]
Therefore, the volume of Kairay's sculpture is approximately 9688 cubic inches.