A solid metallic sphere of radius 4 cm is melted and recast into '4' identical cubes. What is the side of the cube?

Options

A. `7 sqrt(pi/3)` cm
B. `5^3 sqrt(pi/3)`cm
C. `6^3 sqrt (pi/3) cm`
D. `4^3 sqrt(pi/3)` cm

Answer

To find the side of the cube formed by melting a solid metallic sphere, we will follow these steps:

### Step 1: Calculate the volume of the sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Given that the radius \( r = 4 \) cm, we can substitute this value into the formula.

### Step 2: Substitute the radius into the volume formula
Substituting \( r = 4 \) cm into the volume formula:
\[ V = \frac{4}{3} \pi (4)^3 \]
Calculating \( (4)^3 \):
\[ (4)^3 = 64 \]
So, the volume becomes:
\[ V = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

### Step 3: Set the volume of the sphere equal to the volume of the cubes
Since the sphere is melted and recast into 4 identical cubes, the total volume of the cubes will be equal to the volume of the sphere. The volume \( V_c \) of one cube with side length \( A \) is given by:
\[ V_c = A^3 \]
Thus, the total volume of 4 cubes is:
\[ 4 \times A^3 \]
Setting the volume of the sphere equal to the total volume of the cubes:
\[ \frac{256}{3} \pi = 4 A^3 \]

### Step 4: Solve for \( A^3 \)
To isolate \( A^3 \), we divide both sides by 4:
\[ A^3 = \frac{256}{3 \times 4} \pi = \frac{256}{12} \pi = \frac{64}{3} \pi \]

### Step 5: Calculate the side length \( A \)
Now, we take the cube root of both sides to find \( A \):
\[ A = \sqrt[3]{\frac{64}{3} \pi} \]
This can be simplified further:
\[ A = \sqrt[3]{64} \times \sqrt[3]{\frac{\pi}{3}} = 4 \times \sqrt[3]{\frac{\pi}{3}} \]

### Final Answer
Thus, the side of each cube is:
\[ A = 4 \sqrt[3]{\frac{\pi}{3}} \text{ cm} \]