The perimeter of the base ofa right circular cone is 8 cm. If the height of the cone is 21 cm then its volume is :

Options

A. a)`108 pi cm^(3)`
B. b)`(112)/(pi) cm^(3)`
C. c)`112 pi cm^(3)`
D. d)`(108)/(pi) cm^(3)`

Answer

To find the volume of a right circular cone given the perimeter of its base and its height, we can follow these steps:

### Step 1: Find the radius of the base
The perimeter (circumference) of the base of a cone is given by the formula:
\[ C = 2\pi r \]
Given that the perimeter is 8 cm, we can set up the equation:
\[ 2\pi r = 8 \]
To find the radius \( r \), we will isolate \( r \):
\[ r = \frac{8}{2\pi} = \frac{4}{\pi} \text{ cm} \]

### Step 2: Identify the height of the cone
The height \( h \) of the cone is given in the problem as:
\[ h = 21 \text{ cm} \]

### Step 3: Use the volume formula for a cone
The volume \( V \) of a cone is calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \frac{1}{3} \pi \left(\frac{4}{\pi}\right)^2 \times 21 \]

### Step 4: Calculate \( r^2 \)
Calculating \( r^2 \):
\[ r^2 = \left(\frac{4}{\pi}\right)^2 = \frac{16}{\pi^2} \]

### Step 5: Substitute \( r^2 \) into the volume formula
Now substituting \( r^2 \) back into the volume formula:
\[ V = \frac{1}{3} \pi \left(\frac{16}{\pi^2}\right) \times 21 \]

### Step 6: Simplify the expression
Now simplify the expression:
\[ V = \frac{1}{3} \times 21 \times \frac{16}{\pi} = \frac{336}{3\pi} = \frac{112}{\pi} \text{ cm}^3 \]

### Final Answer
Thus, the volume of the cone is:
\[ V = \frac{112}{\pi} \text{ cm}^3 \]

---