The volumes of two cubes are in the ratio 8 : 27. Find the ratio of their surface areas.

Answer

To solve the problem of finding the ratio of the surface areas of two cubes given that their volumes are in the ratio of 8:27, follow these steps:

### Step 1: Understand the relationship between volume and side length of a cube
The volume \( V \) of a cube is given by the formula:
\[ V = A^3 \]
where \( A \) is the length of a side of the cube.

### Step 2: Set up the ratio of the volumes
Given that the volumes of the two cubes are in the ratio:
\[ V_1 : V_2 = 8 : 27 \]
we can express this in terms of their side lengths:
\[ A_1^3 : A_2^3 = 8 : 27 \]

### Step 3: Take the cube root of the volume ratio
To find the ratio of the side lengths \( A_1 \) and \( A_2 \), we take the cube root of both sides:
\[ \frac{A_1}{A_2} = \sqrt[3]{\frac{8}{27}} = \frac{2}{3} \]
Thus, the ratio of the side lengths is:
\[ A_1 : A_2 = 2 : 3 \]

### Step 4: Use the formula for surface area of a cube
The surface area \( S \) of a cube is given by the formula:
\[ S = 6A^2 \]
Thus, the surface areas of the two cubes can be expressed as:
\[ S_1 = 6A_1^2 \quad \text{and} \quad S_2 = 6A_2^2 \]

### Step 5: Set up the ratio of the surface areas
Now, we can set up the ratio of the surface areas:
\[ S_1 : S_2 = 6A_1^2 : 6A_2^2 \]
The \( 6 \) cancels out:
\[ S_1 : S_2 = A_1^2 : A_2^2 \]

### Step 6: Square the ratio of the side lengths
Now we square the ratio of the side lengths:
\[ A_1^2 : A_2^2 = (2^2) : (3^2) = 4 : 9 \]

### Final Answer
Thus, the ratio of the surface areas of the two cubes is:
\[ \text{Surface Area Ratio} = 4 : 9 \]