If a hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that – space of the cube remains unfilled.
Then, the number of marbles that the cube can accomodate is
(a) 142244
(b) 142344
(c) 142444
(d) 142544
(a) Given, edge of the cube = 22 cm
$\therefore \quad$ Volume of the cube $=(22)^{3}=10648 \mathrm{~cm}^{3} \quad\left[\because\right.$ volume of cube $\left.=(\text { side })^{3}\right]$
Also, given diameter of marble $=0.5 \mathrm{~cm}$
$\therefore$ Radius of a marble, $r=\frac{0.5}{2}=0.25 \mathrm{~cm}$ $[\because$ diameter $=2 \times$ radius $]$
Volume of one marble $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \times \frac{22}{7} \times(0.25)^{3}$
$\left[\because\right.$ volume of sphere $\left.=\frac{4}{3} \times \pi \times(\text { radius })^{3}\right]$
$=\frac{1.375}{21}=0.0655 \mathrm{~cm}^{3}$
Filled space of cube $=$ Volume of the cube $\frac{1}{8} \times$ Volume of cube
$=10648-10648 \times \frac{1}{8}$
$=10648 \times \frac{7}{8}=9317 \mathrm{~cm}^{3}$
$\therefore$ Required number of marbles $=\frac{\text { Total space filled by marbles in a cube }}{\text { Volume of one marble }}$
$=\frac{9317}{0.0655}=142244$ (approx)
Hence, the number of marbles that the cube can accomodate is $142244 .$
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