The ratio of the volume of two cubes is 27 :64 then find the ratio of their total surface area

Options

A. 1.1694444444444
B. 3:4
C. 0.38611111111111
D. 3: 8

Answer

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

### Step-by-Step Solution:

1. **Understanding the Volume Ratio**:
The volume of a cube is given by the formula \( V = a^3 \), where \( a \) is the length of a side of the cube. We are given that the ratio of the volumes of two cubes is \( 27:64 \). This can be expressed as:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]

2. **Expressing the Volume in Terms of Side Lengths**:
Let the side lengths of the two cubes be \( a_1 \) and \( a_2 \). Therefore, we can write:
\[ \frac{a_1^3}{a_2^3} = \frac{27}{64} \]

3. **Taking the Cube Root**:
To find the ratio of the side lengths, we take the cube root of both sides:
\[ \frac{a_1}{a_2} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

4. **Finding the Total Surface Area**:
The total surface area \( S \) of a cube is given by the formula \( S = 6a^2 \). Thus, for the two cubes, we have:
\[ S_1 = 6a_1^2 \quad \text{and} \quad S_2 = 6a_2^2 \]

5. **Finding the Ratio of Total Surface Areas**:
The ratio of the total surface areas can be expressed as:
\[ \frac{S_1}{S_2} = \frac{6a_1^2}{6a_2^2} = \frac{a_1^2}{a_2^2} \]

6. **Substituting the Ratio of Side Lengths**:
We already found that \( \frac{a_1}{a_2} = \frac{3}{4} \). Therefore, squaring this ratio gives us:
\[ \frac{a_1^2}{a_2^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

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

### Conclusion:
The ratio of the total surface areas of the two cubes is \( 9:16 \).