Question

Question 8
A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that $\frac{1}{8}$ space of the cube remains unfilled. Then, the number of marbles that the cube can accommodate is

Thinking process- If we divide the total volume filled by marbles in a cube by volume of a marble, then we get the required number of marbles.

(A) 142296
(B) 142396
(C) 142496
(D) 142596

Answer 1

Option (A) is correct.
Given , edge of the cube = 22 cm
$\therefore$ Volume of the cube
$={22}^3$
$=10648c{m}^3$
[Volume of cube = $sid{e}^3$]
Also, given diameter of marble = 0.5 cm

$\therefore$ Radius of a marble,
$\Rightarrow r=\frac{0.5}{2}=0.25cm$ [Diameter = $2\times radius]$

Volume of one marble
$=\frac{4}{3}\pi r^3$
$=\frac{4}{3}\times \frac{22}{7}\times (0.25{)}^3$

Volume of sphere
$=\frac{4}{3}\times \pi (radius{)}^3$
$=\frac{1.375}{21}=0.0655c{m}^3$

Filled space of cube = Volume of the cube - [$\frac{1}{8}\times$ Volume of cube ]
$=10648-(10648\times \frac{1}{8}$\\
$=10648\times \frac{7}{8}=9317c{m}^3$

$\therefore$ Required number of marbles
$=\frac{Totalspacefilledbymarblesinacube}{Volumeofonemarble}$
$=\frac{9317}{0.0655}$
$=142244$ ~ (approx.)

Hence, the number of marbles that cube can accommodate is 142244.