If each side of a cube becomes n times, its volume becomes:

Options

A. 3 N times `n^2` times
B. `n^3` times
C. `3n^3` times.
D. NONE

Answer

To solve the problem, we need to understand how the volume of a cube changes when each side of the cube is multiplied by a factor of \( n \).

### Step-by-Step Solution:

1. **Identify the Initial Side Length:**
Let the initial side length of the cube be \( A \).

2. **Calculate the Initial Volume:**
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
Therefore, the initial volume \( V_{\text{initial}} \) is:
\[ V_{\text{initial}} = A^3 \]

3. **Determine the New Side Length:**
According to the problem, each side of the cube becomes \( n \) times the original side length. Thus, the new side length \( A_{\text{final}} \) is:
\[ A_{\text{final}} = n \times A \]

4. **Calculate the Final Volume:**
Using the new side length, we can calculate the final volume \( V_{\text{final}} \):
\[ V_{\text{final}} = (A_{\text{final}})^3 = (n \times A)^3 \]
Expanding this, we get:
\[ V_{\text{final}} = n^3 \times A^3 \]

5. **Relate Final Volume to Initial Volume:**
We know that \( A^3 = V_{\text{initial}} \). Therefore, we can substitute this into our equation for the final volume:
\[ V_{\text{final}} = n^3 \times V_{\text{initial}} \]

6. **Conclusion:**
Thus, if each side of a cube becomes \( n \) times its original length, the volume becomes \( n^3 \) times the initial volume.

### Final Answer:
If each side of a cube becomes \( n \) times, its volume becomes \( n^3 \) times the initial volume.

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