Step 1: Understanding the Concept:
A cube is a three-dimensional solid object bounded by six square faces.
In a cube, all three dimensions—length, width, and height—are equal.
When we scale the linear dimensions of any 3D object, the volume changes according to the third power (cube) of the scaling factor.
Key Formula or Approach:
The volume \( V \) of a cube with side length \( s \) is given by the formula:
\[ V = s \times s \times s = s^3 \]
Step 2: Detailed Explanation:
Let us consider the initial side length of the cube to be \( s_1 \).
The initial volume of the cube, denoted by \( V_1 \), is:
\[ V_1 = (s_1)^3 \]
The problem states that the side length is doubled. Let the new side length be \( s_2 \).
So, \( s_2 = 2 \times s_1 \).
Now, let's calculate the new volume \( V_2 \) of the cube with this new side length:
\[ V_2 = (s_2)^3 \]
Substitute the value of \( s_2 \) into the equation:
\[ V_2 = (2s_1)^3 \]
Using the algebraic property \( (ab)^n = a^n b^n \), we expand the expression:
\[ V_2 = 2^3 \times (s_1)^3 \]
Since \( 2^3 = 2 \times 2 \times 2 = 8 \), the equation becomes:
\[ V_2 = 8 \times (s_1)^3 \]
Since we know that \( (s_1)^3 \) is the original volume \( V_1 \), we can write:
\[ V_2 = 8 \times V_1 \]
This mathematical derivation shows that the volume has increased by a factor of 8.
If the side had been tripled, the volume would have become \( 3^3 = 27 \) times.
If the side had been halved, the volume would have become \( (1/2)^3 = 1/8 \) of the original.
Step 3: Final Answer:
When the side length of a cube is doubled, its volume increases to eight times the original volume.