If h, c, v are respectively the height curved surface area and volume of a right circular cone, then the value of `3pi v h^(3) - c^(2)h^(2) + 9v^(2)` is

Options

A. a)2
B. b)`-1`
C. c)1
D. d)0

Answer

To solve the problem, we need to find the value of the expression \(3\pi v h^3 - c^2 h^2 + 9v^2\) using the formulas for the height \(h\), curved surface area \(c\), and volume \(v\) of a right circular cone.

### Step 1: Identify the formulas
The formulas for the height \(h\), curved surface area \(c\), and volume \(v\) of a right circular cone are:
- Volume \(v = \frac{1}{3} \pi r^2 h\)
- Curved Surface Area \(c = \pi r l\), where \(l = \sqrt{r^2 + h^2}\)

### Step 2: Substitute the volume \(v\)
We can express \(v\) in terms of \(r\) and \(h\):
\[ v = \frac{1}{3} \pi r^2 h \]

### Step 3: Substitute the curved surface area \(c\)
Now, we substitute for \(c\):
\[ c = \pi r l = \pi r \sqrt{r^2 + h^2} \]

### Step 4: Substitute \(v\) and \(c\) into the expression
Now we substitute \(v\) and \(c\) into the expression \(3\pi v h^3 - c^2 h^2 + 9v^2\):
\[ 3\pi v h^3 = 3\pi \left(\frac{1}{3} \pi r^2 h\right) h^3 = \pi^2 r^2 h^4 \]
\[ c^2 = (\pi r \sqrt{r^2 + h^2})^2 = \pi^2 r^2 (r^2 + h^2) \]
\[ c^2 h^2 = \pi^2 r^2 (r^2 + h^2) h^2 = \pi^2 r^2 h^2 (r^2 + h^2) \]
\[ 9v^2 = 9\left(\frac{1}{3} \pi r^2 h\right)^2 = 9 \cdot \frac{1}{9} \pi^2 r^4 h^2 = \pi^2 r^4 h^2 \]

### Step 5: Combine all parts
Now we can combine all parts into the expression:
\[ 3\pi v h^3 - c^2 h^2 + 9v^2 = \pi^2 r^2 h^4 - \pi^2 r^2 h^2 (r^2 + h^2) + \pi^2 r^4 h^2 \]

### Step 6: Simplify the expression
Now we simplify:
\[ = \pi^2 r^2 h^4 - \pi^2 r^2 h^2 r^2 - \pi^2 r^2 h^2 h^2 + \pi^2 r^4 h^2 \]
\[ = \pi^2 r^2 h^4 - \pi^2 r^4 h^2 - \pi^2 r^2 h^4 + \pi^2 r^4 h^2 \]
\[ = 0 \]

### Final Answer
Thus, the value of the expression \(3\pi v h^3 - c^2 h^2 + 9v^2\) is \(0\).