The ratio of volumes of two cubes is 8 : 125. The ratio of their surface areas is

Options

A. `4:25`
B. `2:75`
C. `2:15`
D. `4:15`

Answer

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

### Step 1: Understand the relationship between volume and side length
The volume \( V \) of a cube with side length \( s \) is given by the formula:
\[ V = s^3 \]
Given the ratio of the volumes of two cubes \( C_1 \) and \( C_2 \) is:
\[ \frac{V_1}{V_2} = \frac{8}{125} \]

### Step 2: Find the ratio of the side lengths
Since the volume is proportional to the cube of the side length, we can take the cube root of both sides to find the ratio of the side lengths \( s_1 \) and \( s_2 \):
\[ \frac{s_1}{s_2} = \sqrt[3]{\frac{8}{125}} = \frac{\sqrt[3]{8}}{\sqrt[3]{125}} = \frac{2}{5} \]

### Step 3: Calculate the surface area of the cubes
The surface area \( A \) of a cube is given by the formula:
\[ A = 6s^2 \]
Now, we will find the surface areas for both cubes using their side lengths.

For cube \( C_1 \):
\[ A_1 = 6(s_1^2) = 6\left(\frac{2}{5}s_2\right)^2 = 6\left(\frac{4}{25}s_2^2\right) = \frac{24}{25}s_2^2 \]

For cube \( C_2 \):
\[ A_2 = 6(s_2^2) = 6s_2^2 \]

### Step 4: Find the ratio of the surface areas
Now, we can find the ratio of the surface areas of the two cubes:
\[ \frac{A_1}{A_2} = \frac{\frac{24}{25}s_2^2}{6s_2^2} = \frac{24}{25 \times 6} = \frac{24}{150} = \frac{4}{25} \]

### Final Answer
Thus, the ratio of the surface areas of the two cubes is:
\[ \frac{A_1}{A_2} = \frac{4}{25} \]