Question

A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and $\frac{1}{8}$ space of the cube remains unfilled. Number of marbles required is
(a) 142296
(b) 142396
(c) 142496
(d) 142596

Answer 1

(a) 142296
Since $\frac{1}{8}\mathrm{th}$ of the cube remains unfulfilled,
volume of the cube = $22\times 22\times 22{\mathrm{cm}}^3$

Space filled in the cube$=\left(\frac{7}{8}\times 22\times 22\times 22\right){\text{cm}}^3$
$=\left(7\times 1331\right){\text{cm}}^3$

Radius of each marble$=\frac{0.5}{2}\text{cm}$
$$\begin{gathered} =\frac{5}{20}\text{cm} \\ =\frac{1}{4}\text{cm} \end{gathered}$$

Volume of each marble$=\frac{4}{3}\mathrm{\pi }{r}^3$
$$\begin{gathered} =\left(\frac{4}{3}\times \frac{22}{7}\times \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}\right){\text{cm}}^3 \\ =\left(\frac{11}{24\times 7}\right){\text{cm}}^3 \end{gathered}$$

Therefore, number of marbles required$=\left(\frac{7\times 1331\times 24\times 7}{11}\right)$
= 142296