Question

A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that $\frac{1}{8}$ space of the cube remains unfilled. Then, the number of marbles that the cube can accommodate is

• A. 122444
• B. 144244
• C. 142442
• D. 142244

Answer 1

The correct option is D 142244

Given, edge of the cube = 22 cm
$\therefore$ Volume of the cube
= side${}^3$
$={22}^3$
$=10648c{m}^3$
Also, given diameter of marble = 0.5 cm
$\Rightarrow$ Radius of a marble,
$r=\frac{0.5}{2}=0.25cm$
[Diameter = $2\times radius]$

Volume of one marble
$=\frac{4}{3}\pi r^3$
$=\frac{4}{3}\times \frac{22}{7}\times (0.25{)}^3$
$=\frac{1.375}{21}=0.0655c{m}^3$

Filled space of cube
= Volume of the cube - $(\frac{1}{8}\times$ Volume of cube)
$=10648-(10648\times \frac{1}{8})=10648\times \frac{7}{8}=9317c{m}^3$

If we divide the total volume filled by marbles in a cube by volume of a marble, then we get the required number of marbles.

$\therefore$ Required number of marbles
$=\frac{Totalspacefilledbymarblesinacube}{Volumeofonemarble}=\frac{9317}{0.0655}=142244(approx.)$

Hence, the number of marbles that cube can accommodate is 142244.