Question

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

Answer 1

Step 1. Given measures of the cylinder

The ratio of radii of two cylinders $=2:3$

And

The ratio of heights $=5:3$

Step 2. Finding the ratios of volumes of the cylinder

Let $r_1$,$h_1$ and $r_2$,$h_2$ be the radii and heights of two cylinders respectively.

$\frac{r_1}{r_2}=\frac{2}{3}$and $\frac{h_1}{h_2}=\frac{5}{3}$

Now,

$\frac{Volumeoffirstcylinder}{Volumeofsecondcylinder}=\frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2{\mathrm{h}}_1}{\mathrm{\pi }{\left(r_2\right)}^2{\mathrm{h}}_2}$

$={\left(\frac{r_1}{r_2}\right)}^2\times \left(\frac{h_1}{h_2}\right)$

$={\left(\frac{2}{3}\right)}^2\times \frac{5}{3}$

$=\frac{4}{9}\times \frac{5}{3}=\frac{20}{27}$

Step 3. Finding the ratios of curved surfaces of the cylinder

$\frac{Curvedsurfaceoffirstcylinder}{Curvedsurfaceofsecondcylinder}=\frac{2{\mathrm{\pi r}}_1{\mathrm{h}}_1}{2{\mathrm{\pi r}}_2{\mathrm{h}}_2}$

$$\begin{gathered} =\left(\frac{r_1}{r_2}\right)\times \left(\frac{h_1}{h_2}\right) \\ =\left(\frac{2}{3}\right)\times \left(\frac{5}{3}\right) \\ =\frac{10}{9} \end{gathered}$$

Hence, the ratios of volumes of the cylinder $20:27$ and the ratios of curved surfaces of the cylinder $10:9$ .