Question

The radii of two cylinders are in the ratio $2:3$ and their heights are in the ratio $5:3.$ Calculate the ratio of their volumes and the ratio of their curved surfaces$.$

Answer 1

Step 1: Find the ratio of the volumes of the given cylinders

Ratio of radius of two cylinder$,r_1:r_2=2:3$

Ratio of their heights$,h_1:h_2=5:3$

Let the radius of the first cylinder be$,$ $r_1=2x$ and the second cylinder is $,r_2=3x.$

The height of the first cylinder be $h_1=5y$ and the second cylinder is $h_2=3y.$

$\because$ The volume of the cylinder having radius $r$ and height $h$$={\mathrm{\pi r}}^2\mathrm{h}$

$\therefore$ The ratio of the volume two cylinders $=\frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2{\mathrm{h}}_1}{\mathrm{\pi }{\left({\mathrm{r}}_2\right)}^2{\mathrm{h}}_2}$

$$\begin{gathered} =\frac{{\left(2x\right)}^2\times 5y}{{\left(3x\right)}^2\times 3y} \\ =\frac{20}{27} \end{gathered}$$

Step 2: Find the ratio of the curved surfaces area of the given cylinders

$\because$The curved surface area of the cylinder having radius $r$ and height $h$$=2\mathrm{\pi rh}$

$\therefore$ Ratio of the curved surface area of the cylinder $=\frac{2\pi r_1{h}_1}{2\pi r_2{h}_2}$

$=\frac{r_1{h}_1}{r_2{h}_2}$

$$\begin{gathered} =\frac{2x\times 5y}{3x\times 3y} \\ =\frac{10}{9} \end{gathered}$$

Hence$,$ the ratio of their volumes and the ratio of their curved surfaces are $20:27$ and $10:9$ respectively.