Two cubes have volumes in the ratio 1:64. The ratio of the area of a face of first cube to that of the other is

Options

A. `1:4`
B. `1:8`
C. `1:16`
D. `1:32`

Answer

To solve the problem, we need to find the ratio of the area of a face of the first cube to that of the second cube, given that the volumes of the two cubes are in the ratio of 1:64.

### Step-by-Step Solution:

1. **Understanding the Volume Ratio**:
The volumes of the two cubes are given in the ratio \( V_1 : V_2 = 1 : 64 \).

2. **Expressing Volume in Terms of Edge Lengths**:
The volume \( V \) of a cube is calculated using the formula:
\[ V = a^3 \]
where \( a \) is the length of the edge of the cube. Therefore, we can express the volumes of the two cubes as:
\[ V_1 = a^3 \quad \text{and} \quad V_2 = A^3 \]
where \( a \) is the edge length of the first cube and \( A \) is the edge length of the second cube.

3. **Setting Up the Volume Ratio**:
From the volume ratio, we have:
\[ \frac{V_1}{V_2} = \frac{a^3}{A^3} = \frac{1}{64} \]

4. **Taking the Cube Root**:
To find the ratio of the edge lengths, we take the cube root of both sides:
\[ \frac{a}{A} = \frac{1}{4} \]

5. **Finding the Ratio of Areas**:
The area of a face of a cube is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Therefore, the areas of the faces of the two cubes are:
\[ \text{Area of first cube} = a^2 \quad \text{and} \quad \text{Area of second cube} = A^2 \]

6. **Setting Up the Area Ratio**:
We need to find the ratio of the areas:
\[ \frac{\text{Area of first cube}}{\text{Area of second cube}} = \frac{a^2}{A^2} \]

7. **Substituting the Edge Length Ratio**:
We can substitute the ratio of the edge lengths into the area ratio:
\[ \frac{a^2}{A^2} = \left(\frac{a}{A}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

8. **Final Answer**:
Thus, the ratio of the area of a face of the first cube to that of the second cube is:
\[ \frac{1}{16} \]