Two cubes have their volumes in the ratio 1 : 27 . The ratio of their surface areas is

Options

A. ` 1 :3`
B. `1:8`
C. `1:9`
D. `1:18`

Explanation

To solve the problem of finding the ratio of the surface areas of two cubes given that their volumes are in the ratio 1:27, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Volume Ratio**:
The volumes of the two cubes are given in the ratio:
\[ V_1 : V_2 = 1 : 27 \]
This means:
\[ \frac{V_1}{V_2} = \frac{1}{27} \]

2. **Volume of a Cube**:
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]
Therefore, for the two cubes, we can express their volumes as:
\[ V_1 = A_1^3 \quad \text{and} \quad V_2 = A_2^3 \]
where \( A_1 \) and \( A_2 \) are the side lengths of the first and second cubes, respectively.

3. **Setting Up the Equation**:
Using the volume expressions:
\[ \frac{A_1^3}{A_2^3} = \frac{1}{27} \]

4. **Finding the Ratio of Side Lengths**:
Taking the cube root of both sides gives:
\[ \frac{A_1}{A_2} = \sqrt[3]{\frac{1}{27}} = \frac{1}{3} \]
Thus, the ratio of the side lengths is:
\[ A_1 : A_2 = 1 : 3 \]

5. **Surface Area of a Cube**:
The surface area \( S \) of a cube with side length \( a \) is given by the formula:
\[ S = 6a^2 \]
Therefore, for the two cubes, we have:
\[ S_1 = 6A_1^2 \quad \text{and} \quad S_2 = 6A_2^2 \]

6. **Setting Up the Surface Area Ratio**:
The ratio of their surface areas can be expressed as:
\[ \frac{S_1}{S_2} = \frac{6A_1^2}{6A_2^2} = \frac{A_1^2}{A_2^2} \]

7. **Substituting the Ratio of Side Lengths**:
Using the ratio of the side lengths:
\[ \frac{S_1}{S_2} = \left(\frac{A_1}{A_2}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

8. **Final Answer**:
Therefore, the ratio of the surface areas of the two cubes is:
\[ S_1 : S_2 = 1 : 9 \]

### Conclusion:
The final answer is that the ratio of the surface areas of the two cubes is \( 1 : 9 \).