A unit radius sphere is inscribed in a maximal volume solid of revolution, and a cylinder. Which of these solids has a larger radius for its base circle?
Using the Pythagorean theorem, $R_{k}^{2}=1-\left(m_{k}-1\right)^{2}$, and $R_{h}^{2}=1-\frac{m_{h}^{2}}{4}$, where $R_{k}$ and $R_{h}$ denote the radii of the base circles of the cone and the cylinder, and $m_{k}$ and $m_{h}$ denote the heights. By differentiating the expressions for the volumes, the maximum points are found to be $m_{k}=\frac{4}{3}$, and $m_{h}^{2}=\frac{4}{3}$, from which $R_{k}=\frac{\sqrt{8}}{3}, R_{h}=\frac{\sqrt{6}}{3}$: the base circle of the cone is larger. (Many started from the incorrect assumption that the volume of the cone or cylinder is exactly maximized when the area of their axial cross-section is.)





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