A metal cube of edge 18 cm is melted to form three smaller cubes. which are unequal in dimensions. If the edges of two smaller cubes are 9 cm and 15 cm, what is the surface area in (`"cm"^2`) of the third smaller cube?

Options

A. 1944
B. 864
C. 1728
D. 486

Answer

To solve the problem step by step, we need to find the surface area of the third smaller cube formed after melting a larger cube.

### 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 18 cm:
\[ V = 18^3 = 5832 \text{ cm}^3 \]

### Step 2: Calculate the volumes of the two smaller cubes.
For the first smaller cube with an edge of 9 cm:
\[ V_1 = 9^3 = 729 \text{ cm}^3 \]

For the second smaller cube with an edge of 15 cm:
\[ V_2 = 15^3 = 3375 \text{ cm}^3 \]

### Step 3: Calculate the volume of the third smaller cube.
Since the original cube is melted to form the three smaller cubes, the volume of the third cube \( V_3 \) can be calculated as:
\[ V_3 = V - (V_1 + V_2) \]
Substituting the values we found:
\[ V_3 = 5832 - (729 + 3375) = 5832 - 4104 = 1728 \text{ cm}^3 \]

### Step 4: Calculate the edge length of the third smaller cube.
Let the edge length of the third cube be \( x \). The volume of the cube can also be expressed as:
\[ V_3 = x^3 \]
Setting the two expressions for \( V_3 \) equal gives:
\[ x^3 = 1728 \]
To find \( x \), we take the cube root:
\[ x = \sqrt[3]{1728} = 12 \text{ cm} \]

### Step 5: Calculate the surface area of the third smaller cube.
The surface area \( A \) of a cube is given by:
\[ A = 6a^2 \]
Substituting \( a = 12 \) cm:
\[ A = 6 \times 12^2 = 6 \times 144 = 864 \text{ cm}^2 \]

### Final Answer:
The surface area of the third smaller cube is \( 864 \text{ cm}^2 \).

---