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. $142296$
• B. $142396$
• C. $142496$
• D. $142596$

Answer 1

The correct option is A $142296$

Volume of hollow cube $={\left(22\right)}^3$

Radius of marble $=\frac{0.5}{2}=\frac{1}{4}$

Volume of each marble $=4\pi r^3=\frac{4}{3}\times \frac{22}{7}\times \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}$

$=\frac{11}{168}{cm}^3$

Space of cube occupied by marbles $=V-\frac{1}{8}V$

$=\frac{7V}{8}$

$\therefore$ Number of marbles $=\frac{\frac{7V}{8}}{\frac{11}{168}}=\frac{\frac{7}{8}\times 22\times 22\times 22}{\frac{11}{168}}=142,296$