Find the rate of change of the volume of a sphere with respect to its diameter.

Options

A. `(pid^2)/2` cubic units.

B. `2(pid^2)` cubic units.

C. `(pid^2)/4` cubic units.

D. None of these

Explanation

To find the rate of change of the volume of a sphere with respect to its diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Relate Diameter to Radius**:
The diameter \( D \) of the sphere is related to the radius by:
\[ D = 2r \quad \Rightarrow \quad r = \frac{D}{2} \]

3. **Substitute Radius in Volume Formula**:
Substitute \( r = \frac{D}{2} \) into the volume formula:
\[ V = \frac{4}{3} \pi \left(\frac{D}{2}\right)^3 \]
Simplifying this gives:
\[ V = \frac{4}{3} \pi \cdot \frac{D^3}{8} = \frac{\pi D^3}{6} \]

4. **Differentiate Volume with Respect to Diameter**:
Now, we differentiate \( V \) with respect to \( D \):
\[ \frac{dV}{dD} = \frac{d}{dD}\left(\frac{\pi D^3}{6}\right) \]
Using the power rule of differentiation:
\[ \frac{dV}{dD} = \frac{\pi}{6} \cdot 3D^2 = \frac{\pi D^2}{2} \]

5. **Conclusion**:
Thus, the rate of change of the volume of the sphere with respect to its diameter is:
\[ \frac{dV}{dD} = \frac{\pi D^2}{2} \]