Two cubes have their volumes in the ratio 27 : 64. The ratio of their surface areas is
दो घनों का घनफल 27:64 के अनुपात में है। उनके पृष्ठीय क्षेत्रफलों का अनुपात है

Options

A. 9 : 25
B. 16 : 25
C. 9 : 16
D. 4 : 9

Explanation

To solve the problem of finding the ratio of the surface areas of two cubes given their volumes in the ratio of 27:64, we can follow these steps:

### Step-by-Step Solution

1. **Understand the relationship between volume and side length of a cube**:
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]

2. **Let the side lengths of the two cubes be \( A \) and \( B \)**:
According to the problem, the volumes of the two cubes are in the ratio:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]
This implies:
\[ \frac{A^3}{B^3} = \frac{27}{64} \]

3. **Take the cube root of both sides**:
To find the ratio of the side lengths \( A \) and \( B \), we take the cube root of the volumes:
\[ \frac{A}{B} = \sqrt[3]{\frac{27}{64}} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

4. **Find the ratio of the surface areas**:
The surface area \( S \) of a cube with side length \( a \) is given by the formula:
\[ S = 6a^2 \]
Therefore, the ratio of the surface areas \( S_1 \) and \( S_2 \) of the two cubes is:
\[ \frac{S_1}{S_2} = \frac{6A^2}{6B^2} = \frac{A^2}{B^2} \]

5. **Substitute the ratio of the sides into the surface area ratio**:
We already found that \( \frac{A}{B} = \frac{3}{4} \). Now we square this ratio to find the ratio of the surface areas:
\[ \frac{A^2}{B^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

6. **Conclusion**:
Thus, the ratio of the surface areas of the two cubes is:
\[ \frac{S_1}{S_2} = \frac{9}{16} \]

### Final Answer:
The ratio of their surface areas is \( 9:16 \).