Three cubes, whose edges are x cm, 8cm and 10cm respectively, are melted and recasted into a single cube of edge 12cm. Find 'x'

Answer

To solve the problem, we need to find the value of \( x \) given that three cubes with edges \( x \) cm, 8 cm, and 10 cm are melted and recast into a single cube with an edge of 12 cm.

### Step-by-step Solution:

1. **Calculate the Volume of Each Cube:**
- The volume of the first cube (edge \( x \) cm) is given by:
\[ V_1 = x^3 \text{ cm}^3 \]
- The volume of the second cube (edge 8 cm) is:
\[ V_2 = 8^3 = 512 \text{ cm}^3 \]
- The volume of the third cube (edge 10 cm) is:
\[ V_3 = 10^3 = 1000 \text{ cm}^3 \]

2. **Calculate the Total Volume of the Three Cubes:**
- The total volume of the three cubes is:
\[ V_{total} = V_1 + V_2 + V_3 = x^3 + 512 + 1000 \]
- Simplifying this gives:
\[ V_{total} = x^3 + 1512 \text{ cm}^3 \]

3. **Calculate the Volume of the New Cube:**
- The volume of the new cube (edge 12 cm) is:
\[ V_{new} = 12^3 = 1728 \text{ cm}^3 \]

4. **Set the Total Volume Equal to the Volume of the New Cube:**
- According to the problem, the total volume of the three cubes equals the volume of the new cube:
\[ x^3 + 1512 = 1728 \]

5. **Solve for \( x^3 \):**
- Rearranging the equation gives:
\[ x^3 = 1728 - 1512 \]
- Calculating the right side:
\[ x^3 = 216 \]

6. **Find \( x \):**
- To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{216} \]
- Since \( 6^3 = 216 \), we find:
\[ x = 6 \text{ cm} \]

### Final Answer:
The value of \( x \) is \( 6 \) cm.