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 volume of two cylinders:

Let, $r,h$ be the radius and height of the first cylinder.

Let $R,H$ be the radius and height of the second cylinder.

Given that, $\frac{r}{R}=\frac{2}{3},\frac{h}{H}=\frac{5}{3}$

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

Therefore, the ratio of the volume of two cylinders $=\frac{{\mathrm{\pi r}}^2\mathrm{h}}{{\mathrm{\pi R}}^2\mathrm{H}}$

$$\begin{gathered} ={\left(\frac{r}{R}\right)}^2\left(\frac{h}{H}\right) \\ ={\left(\frac{2}{3}\right)}^2\left(\frac{5}{3}\right) \\ =\frac{20}{27} \end{gathered}$$

Step 2: Find the ratio of curved surface area

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

So, ratio $=\frac{2\mathrm{\pi rh}}{2\mathrm{\pi RH}}$

$$\begin{gathered} =\left(\frac{r}{R}\right)\left(\frac{h}{H}\right) \\ =\left(\frac{2}{3}\right)\left(\frac{5}{3}\right) \\ =\frac{10}{9} \end{gathered}$$

Therefore, the ratio of volume and curved surface area of two given cylinder is $\frac{20}{27},\frac{10}{9}$ respectively.