Three solid cubes of edges 6 cm , 10 cm and x cm are melted to form a single cube of edge 12 cm. Find the value of x.

Answer

To solve the problem of finding the value of \( x \) when three solid cubes of edges 6 cm, 10 cm, and \( x \) cm are melted to form a single cube of edge 12 cm, we can follow these steps:

### Step 1: Calculate the volume of each cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{edge}^3 \]
- For the cube with edge 6 cm:
\[ V_1 = 6^3 = 216 \text{ cm}^3 \]
- For the cube with edge 10 cm:
\[ V_2 = 10^3 = 1000 \text{ cm}^3 \]
- For the cube with edge \( x \) cm:
\[ V_3 = x^3 \text{ cm}^3 \]

### Step 2: Calculate the volume of the larger cube
The volume of the larger cube with edge 12 cm is:
\[ V = 12^3 = 1728 \text{ cm}^3 \]

### Step 3: Set up the equation
Since the three smaller cubes are melted to form the larger cube, the total volume of the smaller cubes equals the volume of the larger cube:
\[ V_1 + V_2 + V_3 = V \]
Substituting the volumes we calculated:
\[ 216 + 1000 + x^3 = 1728 \]

### Step 4: Simplify the equation
Combine the known volumes:
\[ 1216 + x^3 = 1728 \]

### Step 5: Solve for \( x^3 \)
Subtract 1216 from both sides:
\[ x^3 = 1728 - 1216 \]
Calculating the right side:
\[ x^3 = 512 \]

### Step 6: Find \( x \)
To find \( x \), take the cube root of both sides:
\[ x = \sqrt[3]{512} \]
Calculating the cube root:
\[ x = 8 \text{ cm} \]

### Final Answer
The value of \( x \) is \( 8 \text{ cm} \).

---