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Question

A cone, a hemisphere and a cylinder stand on a equal bases and have the same height. Show that their volume are in the ratio $$1:2:3$$.


Solution

Volume of cone $$=\dfrac{1}{3}\pi{r}^{2}h$$
Volume of hemisphere $$=\dfrac{2}{3}\pi{r}^{3}$$
Volume of cylinder $$=\pi {r}^{2}h$$
Given that cone, hemisphere and cylinder have equal base and same height
That is $$r = h$$
Volume of cone $$:$$ Volume of hemisphere $$:$$ Volume of cylinder  $$=\dfrac{1}{3}\pi{r}^{2}h:  \dfrac{2}{3}\pi{r}^{3}: \pi {r}^{2}h$$
$$=\dfrac{1}{3}\pi{r}^{3}:  \dfrac{2}{3}\pi{r}^{3}: \pi {r}^{3}$$
$$=\dfrac{1}{3}:  \dfrac{2}{3}: 1$$
$$=1:2:3$$
Hence proved.

Mathematics

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