V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3
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- For the sphere with radius \(2x\):
- V_1 = \frac{4}{3}\pi (2x)^3 = \frac{4}{3}\pi (8x^3) = \frac{32}{3}\pi x^3
- The volume \(V_{\text{hemisphere}}\) of a hemisphere with radius \(r\) is half the volume of a full sphere:
- For the hemisphere with radius \(3x\):
- V_2 = \frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi (27x^3) = 18\pi x^3
- \frac{V_2}{V_1} = \frac{18\pi x^3}{\frac{32}{3}\pi x^3} = \frac{18}{\frac{32}{3}} = 18 \cdot \frac{3}{32} = \frac{54}{32} = \frac{27}{16}
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