Question

The volume of a cube is \[V\]. What is the total length of its edges?
1) \[6{V^{\dfrac{1}{3}}}\]
2) \[8\sqrt V \]
3) \[12{V^{\dfrac{2}{3}}}\]
4) \[12{V^{\dfrac{1}{3}}}\]

Answer 1

Hint: We are given the volume of the cube as \[V\] so, we will use the formula of the volume of the cube that is \[V = {\left( {edge} \right)^3}\] where edge represents the side of the cube. We need to find the total length of its edges so, for that we will transform the formula such that the equation comes in terms of volume and gives the value of the edge. Next, multiply the value of the edge with 12.

Complete step-by-step answer:
Consider the volume of a cube as \[V\].
We have to find the total length of the edges of the cube.
As we know that the formula for volume of the cube is \[V = {\left( {edge} \right)^3}\],
Thus, we will use the formula to transform the equation and find the value of the edge in terms of volume.
Thus, we get,
\[ \Rightarrow edge = {\left( V \right)^{\dfrac{1}{3}}}\]
Hence, we get the value of the edge as \[{\left( V \right)^{\dfrac{1}{3}}}\].
Next, we know that there are a total of 12 edges in the cube and to determine the value of the total length of its edges, we will multiply the obtained value of edge by 12.
Thus, we get,
\[
   \Rightarrow {\text{total length}} = 12 \times {\left( V \right)^{\dfrac{1}{3}}} \\
   \Rightarrow {\text{total length}} = 12{V^{\dfrac{1}{3}}} \\
\]
Hence, the total length of the edges of the cube is \[12{V^{\dfrac{1}{3}}}\].
Thus, option D is correct.

Note: Cube is a three-dimensional figure bounded by 6 square faces or sides with three meetings at each vertex. The cube has 6 faces, 12 edges, and 8 vertices in total. The volume of the cube is given by the cube of its edge and the edge length can be found by taking the cube root of the volume.