Answer

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**Hint**: The Volume of the bigger cuboidal box should be divided by the volume of the smaller cubical box. Then the volume of the total number of complete boxes which can be accommodated in the box should be subtracted from the volume of the bigger cuboidal box to obtain the empty space.

**:**

__Complete step-by-step answer__Given information

The dimensions of the cuboidal box are as follows,

Length of the box ,

$

l = 1.2{\text{ m}} \\

l = 1.2 \times 100{\text{ cm}} \\

l{\text{ = 120 cm}} \\

$

Breadth of the box,

\[b = 80{\text{ cm}}\]

Height of the box,

\[h = 50{\text{ cm}}\]

Volume of the bigger cuboidal box is given by,

$\Rightarrow {V_1} = l \times b \times h \cdots \left( 1 \right)$

Substitute the value of $l,b$ and $h$ in equation (1), we get

$

\Rightarrow {V_1} = 120 \times 80 \times 50 \\

\Rightarrow {V_1} = 480000{\text{ c}}{{\text{m}}^3} \\

$

The edge of the cubical box is,

$\Rightarrow a = 7{\text{ cm}}$

The volume of the cube is given by,

$\Rightarrow {V_2} = {a^3} \cdots \left( 2 \right)$

Substitute the value of in equation (2), we get

$

\Rightarrow {V_1} = {\left( 7 \right)^3} \\

\Rightarrow {V_1} = 343{\text{ c}}{{\text{m}}^3} \\

$

The number of smaller cubical boxes that can be accommodated in the bigger cuboidal box is given by,

$\Rightarrow n = \dfrac{{{V_1}}}{{{V_2}}} \cdots \left( 3 \right)$

Substitute the value if ${V_1}$ and ${V_2}$ in equation (3), we get

$

\Rightarrow n = \dfrac{{480000}}{{343}} \\

\Rightarrow n = 1399.417 \\

$

The complete boxes that can be accommodated in the box is, $n = 1399$.

It is quite evident that the cubical box does not fully occupy the cuboidal box. It is because the value of is rounded off to a greatest integer. The part 0.417 corresponds to the empty space in the cuboidal box.

The empty space is given by,

$\Rightarrow E = {V_1} - n{V_2} \cdots \left( 4 \right)$

Substitute the value of ${V_1} = 480000$ ,${V_2} = 343$ and $n = 1399$ in equation (4), we get

$

\Rightarrow E = 480000 - 1399 \times 343 \\

\Rightarrow E = 143{\text{ c}}{{\text{m}}^3} \\

$

Thus, the number of boxes are $n = 1399$ and the empty space is , $E = 143{\text{ c}}{{\text{m}}^3}$ .

**Note**: The important steps are

To calculate the number boxes that can be accommodated is calculated by, by dividing the volume of the bigger box from the volume of the cubical box.

The empty space arises because the small cubical box completely fills the entire bigger cuboidal boxes.

The number of complete boxes is rounded off to the nearest greatest integer.

The greatest integer of x returns the value which is less than or equal to x.

For instance,

$

\left[ {3.4} \right] = 3 \\

\left[ {4.1} \right] = 4 \\

$

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