Question

The volume of a cylinder of radius $r$ is $\frac{1}{4}$ of the volume of a rectangular box with a square base of side length $x$. If the cylinder and the box have equal heights, what is r in terms of $x$?

• A. $\frac{x^2}{2\pi }$
• B. $\frac{\pi }{\sqrt{2x}}$
• C. $\frac{\sqrt{2x}}{\pi }$
• D. $\frac{x}{2\sqrt{\pi }}$

Answer 1

The correct option is D $\frac{x}{2\sqrt{\pi }}$

Let the radius of the cylinder be $r$ and length of both the figure be $h$.
$\therefore$ Volume of the rectangular box $=\text{Base area}\times \text{height}=x^{2}h$
and volume of the cylinder $=\pi r^{2}h$
$\because$ Volume of the cylinder is $\frac{1}{4}$ of the volume of a rectangular box
$\Rightarrow x^{2}h=4\pi r^{2}h\Rightarrow r^2=\frac{x^2}{4\pi }\Rightarrow r=\frac{x}{2\sqrt{\pi }}$
Hence, option $\left(B\right)$ is the correct answer.