Question

The ratio of the curved surface areas of two cylinders is 1 : 3. If the ratio of the heights of the two cylinders is 2 : 3, then the ratio of their volumes is

• A. 1 : 2
• B. 1 : 3
• C. 1 : 6
• D. 2 : 9

Answer 1

The correct option is C 1 : 6

Let the radii of the two cylinders be $r_1$ and $r_2$ and their heights be $h_1$ and $h_2$, respectively.
Also, let the curved surface areas of the two cylinders be $c_1$ and $c_2$.
Now, $h_1:h_2=2:3$ and $c_1:c_2=1:3$.
$\therefore \frac{c_1}{c_2}=\frac{1}{3}$
$\Rightarrow \frac{2\pi r_1{h}_1}{2\pi r_2{h}_2}=\frac{1}{3}$
$\Rightarrow \frac{r_1}{r_2}\times \frac{h_1}{h_2}=\frac{1}{3}$
$\Rightarrow \frac{r_1}{r_2}\times \frac{2}{3}=\frac{1}{3}$
$\Rightarrow \frac{r_1}{r_2}=\frac{1}{2}$
$\therefore$ Ratio of their volumes $=\pi r_1^2{h}_1:\pi r_2^2{h}_2$
$={\left(\frac{r_1}{r_2}\right)}^2\times \frac{h_1}{h_2}$
$={\left(\frac{1}{2}\right)}^2\times \frac{2}{3}$
$=\frac{2}{12}=\frac{1}{6}$
Therefore, the ratio of their volumes is 1 : 6.
Hence, the correct answer is option (c).