Question

The height of sand in a cylindrical-shaped can drops $3$ inches when $1$ cubic foot of sand is poured out. What is the diameter, in inches, of the cylinder?

Answer 1

Let $r$ and $h$ be the radius and height of initial sand can respectivel.
$3\text{inches}=\frac{3}{12}=\frac{1}{4}ft$
$\because$ Difference of volumes of the two can $=1f{t}^3$
$\Rightarrow \pi r^{2}h-\pi r^2(h-\frac{1}{4})=1\Rightarrow \pi \times r^2(h-h+\frac{1}{4})=1\Rightarrow r^2=\frac{4}{\pi }\Rightarrow r^2=\frac{2}{\sqrt{\pi }}$
$\Rightarrow d=2r=\frac{4}{\sqrt{\pi }}ft=\frac{4\times 12}{\sqrt{\pi }}ft=\frac{48}{\sqrt{pi}}\text{inches}$
Hence, option $\left(D\right)$ is the correct answer