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A hollow cube of internal edge $22\ cm$ is filled with spherical marbles of diameter $0.5\ cm$ and it is assumed that $\frac{1}{8}$th space of the cube remains unfilled. Then, find the number of marbles that the cube can accommodate.
Given: A hollow cube of internal edge $22\ cm$ is filled with spherical marbles of diameter $0.5\ cm$ and it is assumed that $\frac{1}{8}$th space of the cube remains unfilled.
To do: To find the number of marbles that the cube can accommodate.
Solution:
Volume of hollow cube $V=( 22)^3$
Radius of marble $=\frac{0.5}{2}=\frac{1}{4}$
Volume of each marble $=4\pi r^3=\frac{4}{3}\times\frac{22}{7}\times( \frac{1}{4})^3$
$=\frac{11}{168}\ cm^3$
Space of cube occupied by marbles $=V−\frac{1}{8}V=\frac{7V}{8}$
$\therefore$ Number of marbles $=\frac{\frac{7V}{8}}{\frac{11}{168}}$
$=\frac{\frac{7}{8}\times22\times22\times22}{\frac{11}{168}}$
$=142, 296$
Thus, number of the marbles is $142, 296$.
Updated on: 10-Oct-2022
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