Question

Volume of a cylinder is in joint variation with square of the length of radius of base and its height. Ratio of radii of bases of two cylinders is $2:3$ and ratio of their heights is $5:4,$ then find the ratio of their volumes. [4 Mark]

Answer 1

Let the volume of the cylinder $=V,$

Radius of the base $=R$ and height $=H$.

As per question, $V\propto R^{2}H\Rightarrow V=k.R^{2}H$ ( where $k\ne 0=$ variation constant). [1 Mark]

Now, the volumes of the cylinders be $V_1\text{and}{V}_2$ and their radii are $2r\text{and}3r$ respectively and heights are $5h\text{and}4h$ respectively, [ $\because$ ratio of radii = $2:3$ and ratio of heights $=5:4$ ] [1 Mark]

$\therefore$ from (1) we get, $V_1=k.(2r{)}^2.5h=20k{r}^{2}h[\because R=2r\text{and}H=5h]$
and $V_2=k.(3r{)}^2.4h=36k{r}^{2}h[\because R=3r\text{and}H=4h]$ [1 Mark]

$\therefore \frac{V_1}{V_2}=\frac{20k{r}^{2}h}{36k{r}^{2}h}=\frac{5}{9}\therefore V_1:V_2=5:9$ [1 Mark]

$\therefore$ The ratio of the volumes of the cylinders is $5:9$.