If h,s V be the height , curved surface area and volume of a cone respectively , then ` ( 3pi Vh^(3) + 9V^(2) -s^(2) h^(2))` is equal to :

Options

A. 0
B. `pi `
C. `(V)/(sh)`
D. `(36)/(V)`

Answer

To solve the problem, we need to analyze the relationship between the height (h), curved surface area (s), and volume (V) of a cone.

### Step-by-Step Solution:

1. **Identify the formulas for volume and curved surface area of a cone:**
- The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
- The curved surface area \( s \) of a cone is given by:
\[ s = \pi r l \]
where \( l \) is the slant height, calculated as:
\[ l = \sqrt{h^2 + r^2} \]

2. **Substituting the volume into the expression:**
- We need to evaluate the expression:
\[ 3\pi V h^3 + 9V^2 - s^2 h^2 \]
- Substitute \( V \) into the expression:
\[ 3\pi \left(\frac{1}{3} \pi r^2 h\right) h^3 + 9\left(\frac{1}{3} \pi r^2 h\right)^2 - s^2 h^2 \]

3. **Simplifying the first term:**
- The first term simplifies as follows:
\[ 3\pi \left(\frac{1}{3} \pi r^2 h\right) h^3 = \pi^2 r^2 h^4 \]

4. **Simplifying the second term:**
- The second term simplifies as follows:
\[ 9\left(\frac{1}{3} \pi r^2 h\right)^2 = 9 \cdot \frac{1}{9} \pi^2 r^4 h^2 = \pi^2 r^4 h^2 \]

5. **Substituting for \( s \):**
- Now substitute \( s \) into the expression:
\[ s = \pi r l = \pi r \sqrt{h^2 + r^2} \]
- Therefore, \( s^2 \) becomes:
\[ s^2 = \pi^2 r^2 (h^2 + r^2) \]

6. **Substituting \( s^2 \) into the expression:**
- Now we have:
\[ \pi^2 r^2 h^4 + \pi^2 r^4 h^2 - \pi^2 r^2 (h^2 + r^2) h^2 \]

7. **Expanding the last term:**
- The last term expands to:
\[ \pi^2 r^2 (h^2 + r^2) h^2 = \pi^2 r^2 h^4 + \pi^2 r^4 h^2 \]

8. **Combining all terms:**
- Now combine all the terms:
\[ \pi^2 r^2 h^4 + \pi^2 r^4 h^2 - (\pi^2 r^2 h^4 + \pi^2 r^4 h^2) = 0 \]

### Final Result:
Thus, the expression simplifies to:
\[ 0 \]

### Conclusion:
The final answer is \( 0 \).