If the ratio of edges of two cubes is 4 : 9, then what is the ratio of the volume of the two cubes?

Options

A. `64 : 729`
B. `16 : 81 `
C. ` 4 : 9 `
D. `2 : 3 `

Answer

To find the ratio of the volumes of two cubes given the ratio of their edges, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We are given the ratio of the edges of two cubes, which is 4:9. We need to find the ratio of their volumes.

2. **Define the Edges**: Let the length of the edge of the first cube be \( A_1 \) and the length of the edge of the second cube be \( A_2 \). According to the problem, we have:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

3. **Volume of a Cube**: The volume \( V \) of a cube is given by the formula:
\[ V = \text{(edge length)}^3 \]
Therefore, the volumes of the two cubes can be expressed as:
\[ V_1 = A_1^3 \quad \text{and} \quad V_2 = A_2^3 \]

4. **Ratio of Volumes**: We want to find the ratio of the volumes \( V_1 \) and \( V_2 \):
\[ \frac{V_1}{V_2} = \frac{A_1^3}{A_2^3} = \left(\frac{A_1}{A_2}\right)^3 \]

5. **Substitute the Ratio of Edges**: Now, substitute the ratio of the edges into the volume ratio:
\[ \frac{V_1}{V_2} = \left(\frac{4}{9}\right)^3 \]

6. **Calculate the Cubes**: Calculate \( \left(\frac{4}{9}\right)^3 \):
\[ \frac{4^3}{9^3} = \frac{64}{729} \]

7. **Final Ratio**: Therefore, the ratio of the volumes of the two cubes is:
\[ \frac{V_1}{V_2} = 64:729 \]

### Conclusion:
The ratio of the volumes of the two cubes is \( 64:729 \).