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 ?

(a) $\frac{24}{\sqrt{\pi }}$ (b) $\frac{48}{\sqrt{\pi }}$

(c) $\frac{32}{\sqrt{\pi }}$ (d) $\frac{48}{\sqrt{\pi }}$

Answer 1

Let h be the height and d be the diameter of a cylinder, then

Radius $=\frac{d}{2},volume=\pi {\left(\frac{d}{2}\right)}^{2}h$

$\therefore Volume=\frac{1}{4}\pi d^{2}h$

Now let us take h =1 foot. Then volume of sand with height of 1 foot becomes $\frac{1}{4}\pi d^2.1$

If height is 3 inches or $\frac{1}{4}ft$. dropped, then

Volume of sand = [$\frac{1}{4}\pi d^2\frac{1}{4}$ ]

$\to 1=\frac{1}{16}\pi d^2\Rightarrow d^2=\frac{16}{\pi }$

$\Rightarrow d=\sqrt{\frac{16}{\pi }}=\frac{4}{\sqrt{\pi }}ft$.

$\therefore d=\frac{4\times 12}{\sqrt{\pi }}$ inches

$\therefore Diameter=\frac{48}{\pi }$ inches