Question

A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that \[\dfrac{1}{8}\] 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

Hint: In this question, we first the volume of the cube with a given edge using the formula \[{{a}^{3}}\]. Then we need to find the volume of the spherical marbles of given diameter using the formula \[\dfrac{4}{3}\pi {{r}^{3}}\]. Now, we need to find the volume which is being occupied by subtracting \[\dfrac{1}{8}\] volume of the cube from the cube volume and then divide it with the volume of marbles to get the number of marbles.

Complete step by step solution:
CUBE:
A cuboid whose length, breadth, height is the same is called a cube.
The volume of a cube of side a is given by
\[\Rightarrow {{a}^{3}}\]
SPHERE:
A sphere is a solid generated by the revolution of a semicircle about its diameter.
The volume of a sphere of radius r is given by
\[\Rightarrow \dfrac{4}{3}\pi {{r}^{3}}\]
Let us assume the volume of the cube as V
Now, given that the edge of the cube is 22 cm
\[a=22\]
Now, the volume of the cube is given by
\[\Rightarrow V={{a}^{3}}\]
Now, on substituting the respective value in the above formula we get,
\[\Rightarrow V={{\left( 22 \right)}^{3}}\]
Now, on further simplification we get,
\[\therefore V=10648c{{m}^{3}}\]
Now, let us find the volume of each of the marble
Given that diameter of the marble is 0.5 cm
As we already know that the relation between the radius and diameter is given by
\[\Rightarrow r=\dfrac{d}{2}\]
Now, on substituting the respective value of diameter we get,
\[\Rightarrow r=\dfrac{0.5}{2}\]
Now, on further simplification we get,
\[\therefore r=0.25cm\]

Let us now find the volume of the marble
\[\Rightarrow \dfrac{4}{3}\pi {{r}^{3}}\]
Now, on substituting the value of radius we get,
\[\Rightarrow \dfrac{4}{3}\pi {{\left( 0.25 \right)}^{3}}\]
Now, on further simplification we get,
\[\Rightarrow \dfrac{11}{168}c{{m}^{3}}\]
Now, the volume occupied by these marbles is given by
\[\Rightarrow V-\dfrac{1}{8}V\]
Now, on further simplification we get,
\[\Rightarrow \dfrac{7}{8}V\]
Now, the number of marbles present is given by
\[\Rightarrow \dfrac{\dfrac{7}{8}V}{\dfrac{11}{168}}\]
Now, on further substituting the respective value we get,
\[\Rightarrow \dfrac{7}{8}\times 10648\times \dfrac{168}{11}\]
Now, on cancelling the common terms we get,
\[\Rightarrow 7\times 121\times 168\]
Now, on further simplification we get,
\[\Rightarrow 142296\]
Hence, the correct option is (a).

Note:
Instead of dividing the volume of the cube being occupied with the volume of marble to get the number of marbles we can also solve it assuming there are some x marbles and then find their volume and then equate it to the volume of the cube being occupied which on simplification gives the value of x.
It is important to note that we can directly divide the complete volume of the cube with the marble volume because only some part of the volume of the cube is being occupied by the marbles. If not considered so then the result will be incorrect.