Two solid right circular cones have the same height. The radii of their bases are $r_{1}$ and $r_{2}$ . They are melted and recast into a cylinder of same height. Show that the radius of the base of the cylinder is $\sqrt{\frac{r_{1}^{2} + r_{2}^{2}}{3}}$ .

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Solution

Let $h$ be the height of the two given cones of base radii $r_{1}$ and $r_{2}$ respectively.
Let $R$ be the radius of the cylinder.
Height of cylinder and cones = $h$ (given)
by given data,
Volume of the cylinder $=$ Sum of the volumes of two cones

$\Rightarrow π R^{2} h = \frac{1}{3} π r_{1}^{2} h + \frac{1}{3} π r_{2}^{2} h$

$\Rightarrow R^{2} = \frac{1}{3} (r_{1}^{2} + r_{2}^{2})$

$\Rightarrow R = \sqrt{\frac{r_{1}^{2} + r_{2}^{2}}{3}}$

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