Question
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.
✓
Answer
Let r be the radius of the base.
and h be the height.
Here, h = r.
Now,
The ratio of their volumes will be
Volume of cone : volume of hemisphere : volume of a cylinder
$\frac{1}{3}\pi\text{r}^2\text{h}:\frac{2}{3}\pi\text{r}^3:\pi\text{r}^2\text{h}$
$\text{V}_1:\text{V}_2:\text{V}_3=\frac{1}{3}\pi\text{r}^3:\frac{2}{3}\pi\text{r}^3:\pi\text{r}^3$
Hence, $\text{V}_1:\text{V}_2:\text{V}_3=1:2:3$
and h be the height.
Here, h = r.
Now,
The ratio of their volumes will be
Volume of cone : volume of hemisphere : volume of a cylinder
$\frac{1}{3}\pi\text{r}^2\text{h}:\frac{2}{3}\pi\text{r}^3:\pi\text{r}^2\text{h}$
$\text{V}_1:\text{V}_2:\text{V}_3=\frac{1}{3}\pi\text{r}^3:\frac{2}{3}\pi\text{r}^3:\pi\text{r}^3$
Hence, $\text{V}_1:\text{V}_2:\text{V}_3=1:2:3$
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