Question

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

• A. $142296$
• B. $142396$
• C. $142496$
• D. $142596$

Answer 1

Answer: (A)

$142296$

Let the spherical marble (as shown in below figure 1) has radius $r$.

Diameter of the marble = $0.5cm$

$\Rightarrow r=\frac{0.5}{2}=0.25cm$

$\Rightarrow r=0.25cm$

Cube (as shown in below figure 2) $l=22cm$

Let $n$ marbles can fill the cube.

$\therefore$ Volume of $n$ marbles = $(1-\frac{1}{8})$ part of volume of cube

$\Rightarrow n.\frac{4}{3}\pi r^3=\frac{7}{8}\times l^3$

$\Rightarrow n=\frac{7l^3}{8}\times \frac{3}{4\pi r^3}=\frac{7\times 3\times 22\times 22\times 22\times 7}{8\times 4\times 22\times 0.25\times 0.25\times 0.25}$

$\Rightarrow n=\frac{7\times 3\times 22\times 22\times 22\times 100\times 100\times 100\times 7}{8\times 4\times 22\times 25\times 25\times 25}$

$=7\times 3\times 22\times 22\times 2\times 7=42\times 484\times 7$

$n=142296$

So, cube can accommodate upto $142296$ marbles so right option is $142296$,

i.e., A other options are more than $142296$. So, cannot accommodate.