Question

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

Answer 1

Let the radii of $2$ cylinders be $r_1$ and $r_2$ respectively and the heights be $h_1$ and $h_2$.

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

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

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

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

Ratio of their surface area $=\frac{10}{9}$

Now, ratio of their volumes $=\frac{\pi r_1^2{h}_1}{\pi r_2^2{h}_2}$

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

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

Ratio of their volumes$=\frac{20}{27}$.