Question

A hollow cube of internal edge $22cm$ is filled with spherical marbles of diameter $0.5cm$ and it is assumed that $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$ 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

$142296$

Edge of cube$=22cm$

Radius of spherical marble $=\frac{0.5}{2}=0.25cm$(Radius is half of the diameter)

Amount of space unfilled in the cube $=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$

According to the given parameters

Volume of cube $={\left(side\right)}^3={\left(22\right)}^3=10648c{m}^3$

Volume of cube that remains unfilled $=\frac{1}{8}\times 10648=1331c{m}^3$

Volume occupied by spherical marbles $=10648-1331=9317c{m}^3$

Volume of 1 spherical marble $\frac{4}{3}{\mathrm{\pi r}}^3=\frac{4}{3}\times \frac{22}{7}\times 0.25\times 0.25\times 0.25=\frac{11}{168}{\mathrm{cm}}^3$

Numbers of spherical marbles, $n=9317÷\frac{11}{168}=142296$

Hence, the number of marbles that the cube can accommodate in the cube is $142296$.