# A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and

Question:

A hollow cube of internal edge $22 \mathrm{~cm}$ is filled with spherical marbles of diameter $0.5 \mathrm{~cm}$ and $\frac{1}{8}$ space of the cube remains unfilled. Number of marbles

required is
(a) 142296
(b) 142396
(c) 142496
(d) 142596

Solution:

(a) 142296

Since $\frac{1}{8}$ th of the cube remains unfulfilled,

volume of the cube $=22 \times 22 \times 22 \mathrm{~cm}^{3}$

Space filled in the cube $=\left(\frac{7}{8} \times 22 \times 22 \times 22\right) \mathrm{cm}^{3}$

$=(7 \times 1331) \mathrm{cm}^{3}$

Radius of each marble $=\frac{0.5}{2} \mathrm{~cm}$

$=\frac{5}{20} \mathrm{~cm}$

$=\frac{1}{4} \mathrm{~cm}$

Volume of each marble $=\frac{4}{3} \pi r^{3}$

$=\left(\frac{4}{3} \times \frac{22}{7} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}\right) \mathrm{cm}^{3}$

$=\left(\frac{11}{24 \times 7}\right) \mathrm{cm}^{3}$

Therefore, number of marbles required $=\left(\frac{7 \times 1331 \times 24 \times 7}{11}\right)$

$=142296$