A metal cube of 9 cm edge is melted and recast into three smaller cubes. If the edge of two of the smaller cubes are 1 cm and 6 cm. Find the edge of the third cube.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the volume of the original cube.
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the edge length of the cube.

For the original cube with an edge of 9 cm:
\[ V = 9^3 = 729 \, \text{cm}^3 \]

### Step 2: Calculate the volume of the first smaller cube.
The edge of the first smaller cube is 1 cm. Using the volume formula:
\[ V_1 = 1^3 = 1 \, \text{cm}^3 \]

### Step 3: Calculate the volume of the second smaller cube.
The edge of the second smaller cube is 6 cm. Using the volume formula:
\[ V_2 = 6^3 = 216 \, \text{cm}^3 \]

### Step 4: Set up the equation for the volume of the third cube.
Let the edge of the third cube be \( x \) cm. The volume of the third cube will be:
\[ V_3 = x^3 \]

According to the problem, the total volume of the three smaller cubes must equal the volume of the original cube:
\[ V = V_1 + V_2 + V_3 \]
Substituting the volumes we calculated:
\[ 729 = 1 + 216 + x^3 \]

### Step 5: Simplify the equation.
Combine the volumes of the first two cubes:
\[ 729 = 217 + x^3 \]

### Step 6: Solve for \( x^3 \).
Subtract 217 from both sides:
\[ x^3 = 729 - 217 \]
\[ x^3 = 512 \]

### Step 7: Find the value of \( x \).
To find \( x \), take the cube root of both sides:
\[ x = \sqrt[3]{512} \]
Calculating the cube root:
\[ x = 8 \, \text{cm} \]

### Conclusion:
The edge of the third cube is \( 8 \, \text{cm} \).

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