Question

The volume of the cube with surface area 384 squares centimeters, is:
A) $216{\text{ c}}{{\text{m}}^3}$
B) $256{\text{ c}}{{\text{m}}^3}$
C) $484{\text{ c}}{{\text{m}}^3}$
D) $512{\text{ c}}{{\text{m}}^3}$

Answer 1

Hint:
In this question, we need to determine the volume of the cube whose surface area has been given as 384 square centimeters. For this we will use the formulae of the surface area and the volume of the cube which is given as ${S_c} = 6{a^2}$ and ${V_c} = {a^3}$.

Complete step by step solution:
Let us consider a cube of side ‘a’ centimeters.

The surface area of the cube is given as 6 times the square of the side of the cube. Mathematically, ${S_c} = 6{a^2}$.
According to the question, the surface area of the cube is 384 squares centimeters. So, substitute this in the formula ${S_c} = 6{a^2}$ to determine the length of the side of the cube.
$
  {S_c} = 6{a^2} \\ \Rightarrow
  384 = 6{a^2} \\ \Rightarrow
  {a^2} = \dfrac{{384}}{6} \\ \Rightarrow
   = 64 - - - - (i) \\
 $
Applying square root to both the sides of the equation (i), we get
$
  {a^2} = 64 \\
  \sqrt {{a^2}} = \sqrt {64} \\ \Rightarrow
  a = 8{\text{ cm}} \\
 $
Hence, the length of the side of the cube is 8 centimeters.
Now, the cube of the length of the side of the cube results in the volume of the cube. Mathematically, ${V_c} = {a^3}$.
So, substitute the side of the cube as 8 centimeters in the formula ${V_c} = {a^3}$ to determine the volume of the cube.
$
  {V_c} = {a^3} \\ \Rightarrow
  {\left( 8 \right)^3} \\ \Rightarrow
  216{\text{ c}}{{\text{m}}^3} \\
 $
Hence, the volume of the cube is 216 cubic centimeters whose surface area is given as 384 squares centimeters.

Option A is correct.

Note:
Observe that, a cube is a three dimensional structure and square is a two dimensional structure. They both are similar in their respective dimensions. The main difference is the dimension. if we’ll give height to a square equal to its other sides then it’ll become a cube.