Question

A cube with an edge length 4 is divided into 8 identical cubes. Calculate the difference between the combined surface area of the 8 smaller cubes and the surface area of the original cube.
(A) 48 (B) 56 (C) 96 (D) 288

Answer 1

Hint: Here we have to think about one basic concept that if a cube melted into one or more than one cube then the total volume of the previous cube is equal to the summation of the all smaller cubes, and using this concept we are gonna solving this particular problem.
Given data,
Edge length of cube = 4 unit
No of smaller identical cubes = 8 nos.

Complete step by step solution:
$\therefore $as we know the volume of cube = ${a^3}{(unit)^3}$ *(Where a = edge length of cube)
Now, volume of cube = ${4^3} = 64uni{t^3}$
We, assume edge length of smaller identical cube = $x$unit
Volume of smaller cube = ${x^3}{(unit)^3}$
$\therefore $now according to hint,
Volume of 8 smaller cube = volume of cube
\[\begin{array}{l}
 \Rightarrow 8 \times {x^3} = 64\\
 \Rightarrow {x^3} = 8\\
 \Rightarrow {x^3} = {2^3}\\
 \Rightarrow x = 2
\end{array}\]
So, the edge length of the smaller cube = 2 units.
Now we have to determine the surface area and have to find the difference between them,
We know, the surface area of cube = $6{a^2}$ *(Where a = edge length of cube)
$\therefore $surface area of cube $ = 6 \times {4^2} = 96uni{t^2}$
$\therefore $surface area of smaller cube $ = 6 \times {2^2} = 24uni{t^2}$
$\therefore $surface area of 8 smaller cube $ = 8 \times 24 = 192uni{t^2}$
Difference, between their surface area = $\left( {192 - 96} \right) = 96uni{t^2}$
The correct option is (C).

Note: - We have to careful about when we are finding the difference between the cube and the smaller cubes, because after finding the surface area of one smaller cube, we have to multiply that with the number of total smaller cubes, then we will get the combined surface area of the smaller cube and the differentiate the quantity. One thing we always have to remember when we are breaking a cube or any object into more than one object then the total combined surface area of a smaller object is always greater than the original one.