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 heightThat 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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