A solid cube with an edge 10 cm is melted to form two equal cubes. The ratio of the edge of the smaller cube to the edge of the bigger cube is

Options

A. `(1/3)^(1/3)`
B. `1/2`
C. `(1/2)^(1/3)`
D. `(1/4)^(1/3)`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the Volume of the Original Cube
The volume \( V \) of a cube can be calculated using the formula:
\[ V = a^3 \]
where \( a \) is the length of the edge of the cube. For the original cube with an edge of 10 cm:
\[ V = 10^3 = 1000 \text{ cm}^3 \]

### Step 2: Determine the Volume of Each Smaller Cube
Since the original cube is melted to form two equal smaller cubes, we need to divide the total volume by 2:
\[ \text{Volume of each smaller cube} = \frac{1000 \text{ cm}^3}{2} = 500 \text{ cm}^3 \]

### Step 3: Calculate the Edge Length of the Smaller Cube
Let the edge length of the smaller cube be \( b \). Using the volume formula for a cube:
\[ b^3 = 500 \]
To find \( b \), we take the cube root of 500:
\[ b = \sqrt[3]{500} \]

### Step 4: Calculate the Edge Length of the Bigger Cube
Since the bigger cube is the original cube, its edge length is still 10 cm.

### Step 5: Find the Ratio of the Edge of the Smaller Cube to the Edge of the Bigger Cube
The ratio of the edge of the smaller cube \( b \) to the edge of the bigger cube (10 cm) is:
\[ \text{Ratio} = \frac{b}{10} \]
Substituting \( b = \sqrt[3]{500} \):
\[ \text{Ratio} = \frac{\sqrt[3]{500}}{10} \]

### Step 6: Simplify the Ratio
To express the ratio in a simpler form, we can calculate \( \sqrt[3]{500} \):
\[ \sqrt[3]{500} \approx 7.937 \]
Thus,
\[ \text{Ratio} \approx \frac{7.937}{10} \approx 0.7937 \]

### Final Answer
The ratio of the edge of the smaller cube to the edge of the bigger cube is approximately \( 0.7937 \).

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