If the volume of two cubes are in the ratio 125 : 1, then what is the ratio of their edges?

Options

A. `5:1`
B. `10:1`
C. `25:1`
D. `125:1`

Answer

To solve the problem, we need to find the ratio of the edges of two cubes given that their volumes are in the ratio of 125:1.

### Step-by-Step Solution:

1. **Understand the Volume of a Cube**:
The volume \( V \) of a cube with edge length \( x \) is given by the formula:
\[ V = x^3 \]

2. **Set Up the Volume Ratio**:
Let the edge lengths of the two cubes be \( x_1 \) and \( x_2 \). According to the problem, the volumes of the cubes are in the ratio:
\[ \frac{V_1}{V_2} = \frac{x_1^3}{x_2^3} = \frac{125}{1} \]

3. **Express the Volume Ratio in Terms of Edge Lengths**:
From the volume ratio, we can write:
\[ \frac{x_1^3}{x_2^3} = \frac{125}{1} \]

4. **Take the Cube Root of Both Sides**:
To find the ratio of the edge lengths, we take the cube root of both sides:
\[ \frac{x_1}{x_2} = \sqrt[3]{\frac{125}{1}} = \sqrt[3]{125} = 5 \]

5. **Write the Ratio of the Edges**:
Thus, the ratio of the edges \( x_1 \) to \( x_2 \) is:
\[ \frac{x_1}{x_2} = \frac{5}{1} \]

6. **Final Answer**:
Therefore, the ratio of their edges is:
\[ 5 : 1 \]