Question

The total length of the edges of a cube is \[96{\text{ cm}}\]. What is the difference in the numbers denoting its total surface area and volume?
(A) \[128\]
(B) \[512\]
(C) \[384\]
(D) \[896\]

Answer 1

Hint: To solve this problem, we have to find the total surface area and volume of the given cube. For this, we will find the length of each edge of the given cube by equating the given sum of all edges to the formula for finding the sum of edges of a cube i.e., \[12 \times \left( {edge} \right)\]. Then we will find the total surface area and volume of the given cube using the formula of total surface area i.e., \[6 \times {\left( {edge} \right)^2}\] and volume i.e., \[{\left( {edge} \right)^3}\] of a cube. Subtracting the obtained total surface area of the cube from the volume of the cube will give the difference in the numbers denoting its total surface area and volume.

Complete step by step answer:
Basic structure of a cube is shown below:

As we know, a cube is made up of \[6\] square faces, \[8\] vertices and $12$ edges.
For a cube, Length \[ = \] Breadth\[ = \] Height i.e., \[L = B = H\]. Therefore, all the edges are of equal length.
Let, length of the edge be \[a\] .
Given, sum of all the edges \[ = 96{\text{ cm}}\]
Since we have considered the length of one edge to be \[a\] , we can write
\[ \Rightarrow 12a = 96{\text{ cm}}\]
On solving,
\[ \Rightarrow a = 8{\text{ cm}}\]
\[\therefore {\text{Length of the edge}} = 8{\text{ cm}}\]
For a cube, total surface area\[ = 6 \times {\left( {edge} \right)^2}\]
\[ = 6 \times {\left( a \right)^2}\]
Putting the value of \[a\] ,
\[ \Rightarrow {\text{Total surface area}} = 6 \times {\left( 8 \right)^2}\]
On solving,
\[ \Rightarrow {\text{Total surface area}} = 384{\text{ c}}{{\text{m}}^2}\]
Now, as volume of the cube\[ = {\left( {edge} \right)^3}\]
\[ = {\left( a \right)^3}\]
On putting the value of \[a\] we get
\[ \Rightarrow {\text{Volume of the cube}} = {\left( 8 \right)^3}\]
On solving,
\[ \Rightarrow {\text{Volume of the cube}} = 512{\text{ c}}{{\text{m}}^3}\]
Therefore, we get
Total surface area of the cube\[ = 384{\text{ c}}{{\text{m}}^2}\] and Volume of the cube\[ = 512{\text{ c}}{{\text{m}}^3}\]
The difference in the numbers denoting its total surface area and volume \[ = 512 - 384\]
\[ = 128\]
Therefore, the difference in numbers denoting its total surface area and volume is 128. Hence, option (A) is correct.

Note:
We have to keep in mind whether we require the total surface area or lateral surface area of the cube. If we require lateral surface area then it will be the sum of all faces except the top and bottom face of the cube. For this, the formula that we will use to find the lateral surface area will be \[\left( {6{a^2} - 2{a^2}} \right)\] i.e., \[4{a^2}\].