Discussion
The maximum volume of a cone that can be carved out of a solid hemisphere of radius r is:
Answer: C
The maximum volume is \(\frac{1}{3}\pi r^{3}\).
The largest cone that fits in a solid hemisphere of radius r has its base on the flat face of radius r and its height equal to r.
\[ V=\frac{1}{3}\pi r^{2}\cdot r=\frac{1}{3}\pi r^{3} \]
\(\frac{2}{3}\pi r^{3}\) is the volume of the hemisphere itself, and \(\frac{4}{3}\pi r^{3}\) is a full sphere.
The cone with the greatest volume has its base on the flat face of the hemisphere and its apex at the top of the curved surface, which makes the height equal to the radius. A hemisphere has volume 2/3 pi r cubed, so the cone is exactly half of it: 1/3 pi r cubed.
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