The ratio of the volumes of two cubes is `729: 1331` .What is the ratio of their total surface areas?

Answer

To find the ratio of the total surface areas of two cubes given the ratio of their volumes, we can follow these steps:

### Step 1: Understand the relationship between volume and edge length of a cube
The volume \( V \) of a cube with edge length \( a \) is given by the formula:
\[ V = a^3 \]
If the volumes of two cubes are in the ratio \( V_1 : V_2 \), then we can express this as:
\[ \frac{V_1}{V_2} = \frac{a_1^3}{a_2^3} \]

### Step 2: Set up the equation with the given volume ratio
From the problem, we know the ratio of the volumes of the two cubes is:
\[ \frac{729}{1331} \]
This can be written as:
\[ \frac{a_1^3}{a_2^3} = \frac{729}{1331} \]

### Step 3: Find the ratio of the edge lengths
To find the ratio of the edge lengths \( a_1 \) and \( a_2 \), we take the cube root of both sides:
\[ \frac{a_1}{a_2} = \sqrt[3]{\frac{729}{1331}} \]
Calculating the cube roots, we find:
\[ 729 = 9^3 \quad \text{and} \quad 1331 = 11^3 \]
Thus:
\[ \frac{a_1}{a_2} = \frac{9}{11} \]

### Step 4: Determine the formula for total surface area
The total surface area \( S \) of a cube with edge length \( a \) is given by:
\[ S = 6a^2 \]
For two cubes, the total surface areas are:
\[ S_1 = 6a_1^2 \quad \text{and} \quad S_2 = 6a_2^2 \]

### Step 5: Set up the ratio of total surface areas
Now, we can find the ratio of their total surface areas:
\[ \frac{S_1}{S_2} = \frac{6a_1^2}{6a_2^2} = \frac{a_1^2}{a_2^2} \]

### Step 6: Substitute the ratio of edge lengths
Substituting the ratio of edge lengths we found earlier:
\[ \frac{S_1}{S_2} = \left(\frac{a_1}{a_2}\right)^2 = \left(\frac{9}{11}\right)^2 = \frac{81}{121} \]

### Final Answer
Thus, the ratio of the total surface areas of the two cubes is:
\[ \frac{81}{121} \]

---