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

• A. $22:23$
• B. $20:27$
• C. $25:29$
• D. $16:19$

Answer 1

The correct option is B

$20:27$

Step $1:$It is given that$,$

Ratio of radii of two cylinder is $2:3$

let radius of first cylinder $\left(r_1\right)=2x$

radius of second cylinder $\left(r_2\right)=3x$

Ratio of their height is $5:3$

Let the height of first cylinder $\left(h_1\right)=5y$

The height of second cylinder $\left(h_2\right)=3y$

Step $2:$Volume of Cylinder $V={\mathrm{\pi r}}^2\mathrm{h}$

$\frac{V_1}{V_2}=\frac{{{\mathrm{\pi r}}_1}^2{\mathrm{h}}_1}{{{\mathrm{\pi r}}_2}^2{\mathrm{h}}_2}$

$\frac{V_1}{V_2}=\frac{{r_1}^2{\mathrm{h}}_1}{{r_2}^2{\mathrm{h}}_2}$

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

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

$=\left(\frac{4}{9}\right)\times \frac{5}{3}$

$=\frac{20}{27}$

Hence$,$ the ratio of their volumes is $20:27$

Therefore, option $\left(B\right)$ is correct.