A metal cube of edge 12 cm, is melted and casted into three small cubes. If the edges of two small cubes be 6 cm and 8 cm, then the edge of the third small cube is

Options

A. 9 cm
B. 25 cm
C. 20 cm
D. 10 cm

Answer

To solve the problem, we need to find the edge of the third small cube after melting a larger cube of edge 12 cm into three smaller cubes with known edges of 6 cm and 8 cm.

### Step-by-Step Solution:

1. **Calculate the Volume of the Larger Cube:**
The volume \( V \) of a cube is given by the formula:
\[ V = \text{edge}^3 \]
For the larger cube with an edge of 12 cm:
\[ V = 12^3 = 1728 \, \text{cm}^3 \]

**Hint:** Remember to use the formula for the volume of a cube, which is the edge length raised to the power of three.

2. **Calculate the Volume of the First Small Cube:**
The first small cube has an edge of 6 cm:
\[ V_1 = 6^3 = 216 \, \text{cm}^3 \]

**Hint:** Use the same volume formula for the first small cube.

3. **Calculate the Volume of the Second Small Cube:**
The second small cube has an edge of 8 cm:
\[ V_2 = 8^3 = 512 \, \text{cm}^3 \]

**Hint:** Again, apply the volume formula for the second small cube.

4. **Set Up the Equation for the Volume of the Third Cube:**
Let the edge of the third small cube be \( x \). Its volume will be \( x^3 \). According to the problem, the total volume of the three small cubes must equal the volume of the larger cube:
\[ V_1 + V_2 + V_3 = V \]
Substituting the known volumes:
\[ 216 + 512 + x^3 = 1728 \]

**Hint:** Ensure that you correctly set up the equation based on the total volume.

5. **Combine and Solve for \( x^3 \):**
First, add the volumes of the two small cubes:
\[ 216 + 512 = 728 \]
Now, substitute this back into the equation:
\[ 728 + x^3 = 1728 \]
To find \( x^3 \), subtract 728 from both sides:
\[ x^3 = 1728 - 728 = 1000 \]

**Hint:** Perform the arithmetic carefully to isolate \( x^3 \).

6. **Find the Edge Length of the Third Cube:**
Now, take the cube root of both sides to find \( x \):
\[ x = \sqrt[3]{1000} \]
Since \( 1000 = 10^3 \):
\[ x = 10 \, \text{cm} \]

**Hint:** Remember that the cube root of a number gives you the edge length of the cube.

### Final Answer:
The edge of the third small cube is **10 cm**.