A cube of 384 `cm^(2)` surface area is melt to make x number of small cubes each of 96 `mm^(2)` surface area. The value of x is

Options

A. 80000
B. 8
C. 8000
D. 800

Answer

To solve the problem, we need to find the number of small cubes that can be made from a larger cube with a given surface area. Here’s the step-by-step solution:

### Step 1: Calculate the side length of the larger cube
The surface area (SA) of a cube is given by the formula:
\[ SA = 6A^2 \]
where \( A \) is the side length of the cube.

Given that the surface area of the larger cube is \( 384 \, cm^2 \):
\[ 6A^2 = 384 \]

### Step 2: Solve for \( A^2 \)
To find \( A^2 \), divide both sides by 6:
\[ A^2 = \frac{384}{6} = 64 \]

### Step 3: Calculate the side length \( A \)
Now, take the square root of \( A^2 \) to find \( A \):
\[ A = \sqrt{64} = 8 \, cm \]

### Step 4: Calculate the volume of the larger cube
The volume (V) of a cube is given by:
\[ V = A^3 \]
Substituting the value of \( A \):
\[ V = 8^3 = 512 \, cm^3 \]

### Step 5: Convert the volume of the larger cube to mm³
Since \( 1 \, cm = 10 \, mm \), we convert the volume from cm³ to mm³:
\[ 1 \, cm^3 = 1000 \, mm^3 \]
Thus,
\[ 512 \, cm^3 = 512 \times 1000 = 512000 \, mm^3 \]

### Step 6: Calculate the side length of the smaller cube
The surface area of the smaller cube is given as \( 96 \, mm^2 \). Using the surface area formula:
\[ 6x^2 = 96 \]

### Step 7: Solve for \( x^2 \)
Divide both sides by 6:
\[ x^2 = \frac{96}{6} = 16 \]

### Step 8: Calculate the side length \( x \)
Taking the square root gives:
\[ x = \sqrt{16} = 4 \, mm \]

### Step 9: Calculate the volume of the smaller cube
Using the volume formula:
\[ V = x^3 \]
Substituting the value of \( x \):
\[ V = 4^3 = 64 \, mm^3 \]

### Step 10: Calculate the number of smaller cubes
To find the number of smaller cubes \( x \), divide the volume of the larger cube by the volume of the smaller cube:
\[ x = \frac{512000 \, mm^3}{64 \, mm^3} = 8000 \]

### Final Answer
The value of \( x \) is \( 8000 \).

---