Question

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

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

Answer 1

The correct option is B $\left(\frac{1}{\sqrt{2}}\right)$

Volume of a cylinder is given by: $V_1=\pi \times r_1^2\times h_1$
where, $r_1=$ radius of the cylinder
Now, height of new cylinder $h_2=2h_1$ (given)
But, the volume of new cylinder $=V_2=\pi \times r_2^2\times h_2$
where, $r_2=$ radius of the new cylinder
Since, the volume is unchanged,
So, $V_2=V_1$
$\Rightarrow \pi \times r_2^2\times h_2=\pi \times r_1^2\times h_1$

$\Rightarrow r_2^2\times h_2=r_1^2\times h_1$

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

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

$\Rightarrow r_2^2=\frac{\left(r_1^2\right)}{2}$

$\therefore r_2=\frac{\left(r_1\right)}{\sqrt{2}}$

Thus, the radius of base should be multiplied by $\frac{1}{\sqrt{2}}$,
so that the resulting cylinder has the same volume.