The ratio of volumes of 2 cubes is `27:64`. What will be the ratio of the areas of surfaces ?

Options

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

Answer

To find the ratio of the areas of surfaces of two cubes given the ratio of their volumes, we can 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 volumes
Given the ratio of the volumes of two cubes is:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]
Let \( V_1 = a_1^3 \) and \( V_2 = a_2^3 \). Thus, we can write:
\[ \frac{a_1^3}{a_2^3} = \frac{27}{64} \]

### Step 3: Take the cube root of both sides
To find the ratio of the side lengths \( a_1 \) and \( a_2 \), we take the cube root of both sides:
\[ \frac{a_1}{a_2} = \sqrt[3]{\frac{27}{64}} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

### Step 4: Calculate the 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 surface areas
Now, we can find the ratio of the 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 side lengths into the surface area ratio
We already found that:
\[ \frac{a_1}{a_2} = \frac{3}{4} \]
Now, squaring this ratio gives:
\[ \frac{a_1^2}{a_2^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Step 7: Conclusion
Thus, the ratio of the areas of surfaces of the two cubes is:
\[ \frac{S_1}{S_2} = \frac{9}{16} \]

### Final Answer
The ratio of the areas of surfaces of the two cubes is \( 9:16 \).

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