Question

The internal measurements of a box are 20 cm long, 16 cm wide and 24 cm high. How many 4 cm cubes could be put into the box?

Answer 1

Hint: Here, we will first find the volume of the box using the formula of volume of a cuboid. Then we will find the volume of cubes using the formula of volume of a cube. We will then divide the volume of the cuboidal box by the volume of each cubical box to find the number of cubes of the same volume to get the required answer.

Formula used:
We will use the following formulas:
1.Volume of a cuboid \[ = l \times b \times h\], where, \[l\] is the length, \[b\] is the breadth and \[h\] is the height of the box respectively.
2.Volume of this cube \[ = {s^3}\], where \[s\] is the side of the cube.

Complete step-by-step answer:
We will first find the volume of the box with the dimension 20 cm long, 16 cm wide and 24 cm high.
Substituting \[l = 20{\rm{cm}}\], \[b = 16{\rm{cm}}\] and \[h = 24{\rm{cm}}\] in the formula volume of a cuboid \[ = l \times b \times h\], we get
Volume of the box \[ = 20 \times 16 \times 24\]
Multiplying the terms, we get
\[ \Rightarrow \] Volume of the box \[ = 384 \times 20 = 7680{\rm{c}}{{\rm{m}}^3}\]
Now, we are given a cube with a side, \[s = 4{\rm{cm}}\].
Substituting \[s = 4{\rm{cm}}\] in the formula volume of this cube \[ = {s^3}\], we get
Volume of cube \[ = {\left( 4 \right)^3}\]
Applying the exponent on the term, we get
\[ \Rightarrow \] Volume of cube \[ = 64{\rm{c}}{{\rm{m}}^3}\]
Now, we have to find that number of 4 cm cubes that could be put into the given box.
Hence, we will divide the volume of the given box by volume of each cube to get the required number of cubes.
Therefore, number of cubes that can be put in the given box \[ = \dfrac{{7680}}{{64}}\]
Dividing numerator and denominator by 8, we get
\[ \Rightarrow \] Number of cubes that can be put in the given box \[ = \dfrac{{960}}{8} = 120\]
Hence, a total of 120 cubes, each with the side 4 cm, can be put in the given box.
Hence 120 is the required answer.

Note: Here we will divide the volume of the box by the volume of the cube and not the other way round because then we will not get the correct answer. We need to keep in mind that the number of cubes can neither be negative nor be in decimals. Also, we might make a mistake by multiplying both the volumes, this will give the wrong answer. Volume is found out to know the capacity of a three- dimensional object.