Two cubes have volumes in the ratio 27 : 216 . What is the ratio of the area of the face of one cube to that of the other cube ?

Options

A. `1 : 4`
B. `1 : 6`
C. `1 : 9`
D. `1 : 18`

Answer

To solve the problem step by step, we will find the ratio of the areas of the faces of the two cubes based on the given ratio of their volumes.

### 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.
We are given that the volumes of two cubes are in the ratio:
\[ \frac{V_1}{V_2} = \frac{27}{216} \]
Let the side lengths of the cubes be \( x \) and \( y \). Therefore, we can write:
\[ \frac{x^3}{y^3} = \frac{27}{216} \]

### Step 3: Simplify the volume ratio.
We can simplify \( \frac{27}{216} \):
\[ \frac{27}{216} = \frac{1}{8} \]
Thus, we have:
\[ \frac{x^3}{y^3} = \frac{1}{8} \]

### Step 4: Find the ratio of the side lengths.
Taking the cube root of both sides, we find:
\[ \frac{x}{y} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \]

### Step 5: Calculate the ratio of the areas of the faces.
The area \( A \) of one face of a cube is given by:
\[ A = a^2 \]
Thus, the areas of the faces of the two cubes are:
\[ A_1 = x^2 \quad \text{and} \quad A_2 = y^2 \]
Now, we find the ratio of the areas:
\[ \frac{A_1}{A_2} = \frac{x^2}{y^2} \]

### Step 6: Substitute the ratio of the side lengths.
Using the ratio \( \frac{x}{y} = \frac{1}{2} \):
\[ \frac{x^2}{y^2} = \left(\frac{x}{y}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Conclusion:
The ratio of the area of the face of one cube to that of the other cube is:
\[ \frac{A_1}{A_2} = \frac{1}{4} \]

### Final Answer:
The ratio of the area of the face of one cube to that of the other cube is \( 1 : 4 \).

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