If a cube of maximum possible volume is cut off from a solid sphere of diameter d , then the volume of the remaining (waste ) material of the sphere would be equal to :

Options

A. `(d^(3))/(3)(pi-(d)/(2))`
B. `(d^(3))/(3)((pi)/(2)-(1)/(sqrt(3)))`
C. `(d^(2))/(4)(sqrt(2)-pi)`
D. none of these

Answer

To find the volume of the remaining material (waste) after cutting a cube of maximum possible volume from a solid sphere of diameter \( d \), we can follow these steps:

### Step 1: Determine the radius of the sphere
The diameter of the sphere is given as \( d \). Therefore, the radius \( r \) of the sphere is:
\[ r = \frac{d}{2} \]

### Step 2: Calculate the volume of the sphere
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi r^3 \]
Substituting the value of \( r \):
\[ V_s = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{4}{3} \pi \left(\frac{d^3}{8}\right) = \frac{\pi d^3}{6} \]

### Step 3: Determine the side length of the cube
The maximum cube that can fit inside the sphere will have its diagonal equal to the diameter of the sphere. The diagonal \( D \) of a cube with side length \( a \) is given by:
\[ D = a \sqrt{3} \]
Setting this equal to the diameter of the sphere:
\[ a \sqrt{3} = d \implies a = \frac{d}{\sqrt{3}} \]

### Step 4: Calculate the volume of the cube
The volume \( V_c \) of the cube is given by:
\[ V_c = a^3 = \left(\frac{d}{\sqrt{3}}\right)^3 = \frac{d^3}{3\sqrt{3}} \]

### Step 5: Calculate the volume of the remaining material
The volume of the remaining material (waste) after cutting out the cube from the sphere is calculated by subtracting the volume of the cube from the volume of the sphere:
\[ V_{\text{remaining}} = V_s - V_c = \frac{\pi d^3}{6} - \frac{d^3}{3\sqrt{3}} \]

### Step 6: Simplify the expression
To combine the two volumes, we need a common denominator. The common denominator for \( 6 \) and \( 3\sqrt{3} \) is \( 6\sqrt{3} \):
\[ V_{\text{remaining}} = \frac{\pi d^3 \sqrt{3}}{6\sqrt{3}} - \frac{2d^3}{6\sqrt{3}} = \frac{\pi d^3 \sqrt{3} - 2d^3}{6\sqrt{3}} \]

### Final Answer
Thus, the volume of the remaining material of the sphere is:
\[ V_{\text{remaining}} = \frac{d^3(\pi \sqrt{3} - 2)}{6\sqrt{3}} \]