Question

If the height of a cylinder is doubled, by what number must the radius of the base be multiplied so that the resulting cylinder has the same volume as the original cylinder?

• A. $2$
• B. $\frac{1}{\sqrt{2}}$
• C. $4$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{\sqrt{2}}$

Let $r_1,h_1$ be the radius and height of original cylinders and $r_2,h_2$ be the radius and height of new cylinders, respectively
According to the question,
$h_2=2h_1$
$\because$ Volumes remains same
$\Rightarrow \pi r_1^2{h}_1=\pi r_2^2{h}_2\Rightarrow {\left(\frac{r_1}{r_2}\right)}^2\times \frac{h_1}{h_2}=1\Rightarrow {\left(\frac{r_1}{r_2}\right)}^2\times \frac{1}{2}=1\Rightarrow \frac{r_1}{r_2}=\frac{\sqrt{2}}{1}\Rightarrow r_2=\frac{r_1}{\sqrt{2}}$
$\therefore$ The radius should be multiplied by $\frac{1}{\sqrt{2}}$ so that the resulting cylinder has the same volume.
Hence, option $\left(B\right)$ is the correct answer.