The ratio of volumes of two cubes is 8:27. What is the ratio of surface area of these cubes respectively?

Options

A. `2:3`
B. `4:9`
C. `8:19`
D. `9:4`

Answer

To solve the problem, we need to find the ratio of the surface areas of two cubes given the ratio of their volumes.

### Step-by-Step Solution:

1. **Understand the relationship between volume and side length of a cube:**
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
If the volumes of the two cubes are in the ratio \( 8:27 \), we can express this as:
\[ \frac{V_1}{V_2} = \frac{8}{27} \]

2. **Set up the equation for the sides of the cubes:**
Let \( s_1 \) be the side length of the first cube and \( s_2 \) be the side length of the second cube. Then we have:
\[ \frac{s_1^3}{s_2^3} = \frac{8}{27} \]

3. **Take the cube root of both sides:**
To find the ratio of the side lengths, we take the cube root:
\[ \frac{s_1}{s_2} = \sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \]

4. **Find the ratio of the surface areas:**
The surface area \( A \) of a cube is given by the formula:
\[ A = 6 \times \text{side}^2 \]
Therefore, the ratio of the surface areas of the two cubes is:
\[ \frac{A_1}{A_2} = \frac{6 \times s_1^2}{6 \times s_2^2} = \frac{s_1^2}{s_2^2} \]

5. **Substituting the ratio of the sides:**
We already found that \( \frac{s_1}{s_2} = \frac{2}{3} \). Now we square this ratio:
\[ \frac{s_1^2}{s_2^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

6. **Final ratio of surface areas:**
Thus, the ratio of the surface areas of the two cubes is:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

### Conclusion:
The ratio of the surface areas of the two cubes is \( 4:9 \).

---