The ratio of volume of two cubes is `27 : 64`, then ratio of their total surface area is:

Options

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

Answer

To find the ratio of the total surface area of two cubes when the ratio of their volumes is given as \(27:64\), we can follow these steps:

### Step 1: Understand the relationship between volume and side length
The volume \(V\) of a cube with side length \(A\) is given by the formula:
\[ V = A^3 \]
If the volumes of two cubes are \(V_1\) and \(V_2\), and their side lengths are \(A_1\) and \(A_2\), then:
\[ \frac{V_1}{V_2} = \frac{A_1^3}{A_2^3} \]

### Step 2: Set up the equation based on the given ratio
We know from the problem that:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]
This can be rewritten as:
\[ \frac{A_1^3}{A_2^3} = \frac{27}{64} \]

### Step 3: Take the cube root of both sides
To find the ratio of the side lengths, we take the cube root of both sides:
\[ \frac{A_1}{A_2} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

### Step 4: Calculate the total surface area
The total surface area \(S\) of a cube is given by the formula:
\[ S = 6A^2 \]
Thus, the ratio of the total surface areas of the two cubes is:
\[ \frac{S_1}{S_2} = \frac{6A_1^2}{6A_2^2} = \frac{A_1^2}{A_2^2} \]

### Step 5: Substitute the ratio of side lengths
Now, substituting the ratio of the side lengths we found earlier:
\[ \frac{S_1}{S_2} = \frac{(A_1/A_2)^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Conclusion
Therefore, the ratio of their total surface areas is:
\[ \frac{S_1}{S_2} = \frac{9}{16} \]