A cube of maximum possible volume is removed from a solif wooden sphere of radius 6 cm. The side of the cube is :

Options

A. `3sqrt(3) cm`
B. `4sqrt(3) cm`
C. `2sqrt(3) cm`
D. `6sqrt(3) cm`

Answer

To solve the problem of finding the side of the cube that can be removed from a solid wooden sphere of radius 6 cm, we can follow these steps:

### Step 1: Understand the relationship between the cube and the sphere
When a cube is inscribed in a sphere, the diagonal of the cube is equal to the diameter of the sphere.

### Step 2: Calculate the diameter of the sphere
The radius of the sphere is given as 6 cm. Therefore, the diameter \( D \) of the sphere can be calculated as:
\[ D = 2 \times \text{radius} = 2 \times 6 = 12 \text{ cm} \]

### Step 3: Relate the diagonal of the cube to its side length
Let the side length of the cube be \( A \) cm. The formula for the diagonal \( d \) of a cube in terms of its side length is:
\[ d = A\sqrt{3} \]
Since the diagonal of the cube is equal to the diameter of the sphere, we can set up the equation:
\[ A\sqrt{3} = 12 \]

### Step 4: Solve for the side length \( A \)
To find \( A \), we can rearrange the equation:
\[ A = \frac{12}{\sqrt{3}} \]
To simplify this, we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ A = \frac{12\sqrt{3}}{3} = 4\sqrt{3} \text{ cm} \]

### Conclusion
The side length of the cube that can be removed from the sphere is \( 4\sqrt{3} \) cm.