If the volumes of two cubes are in the ratio 27: 64, then the ratio of their total surface areas is

Options

A. 27:64
B. `3:4`
C. `9:16`
D. `3:8`

Answer

To solve the problem, we need to find the ratio of the total surface areas of two cubes given that their volumes are in the ratio of 27:64.

### Step-by-Step Solution:

1. **Understanding the Volume Ratio**:
The volumes of two cubes are given in the ratio:
\[ V_1 : V_2 = 27 : 64 \]

2. **Let the Side Lengths of the Cubes**:
Let the side lengths of the two cubes be \( x \) and \( y \). Therefore, we can express the volumes of the cubes as:
\[ V_1 = x^3 \quad \text{and} \quad V_2 = y^3 \]

3. **Setting Up the Volume Equation**:
From the volume ratio, we have:
\[ \frac{x^3}{y^3} = \frac{27}{64} \]

4. **Taking the Cube Root**:
Taking the cube root of both sides gives us the ratio of the sides:
\[ \frac{x}{y} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

5. **Calculating the Total Surface Area**:
The total surface area \( S \) of a cube is given by the formula:
\[ S = 6 \times (\text{side length})^2 \]
Therefore, the total surface areas of the two cubes can be expressed as:
\[ S_1 = 6x^2 \quad \text{and} \quad S_2 = 6y^2 \]

6. **Finding the Ratio of Surface Areas**:
Now, we need to find the ratio of the surface areas:
\[ \frac{S_1}{S_2} = \frac{6x^2}{6y^2} = \frac{x^2}{y^2} \]

7. **Substituting the Ratio of Sides**:
We already found that \( \frac{x}{y} = \frac{3}{4} \). Therefore,
\[ \frac{x^2}{y^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

8. **Final Ratio of Surface Areas**:
Thus, the ratio of the total surface areas of the two cubes is:
\[ S_1 : S_2 = 9 : 16 \]

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