The perimeter of base of a right circular cone is 132 cm. If the height of the cone is 72 cm, then what is the curved surface area ( in `cm^(2)`) of the cone ?

Options

A. 6600
B. 6336
C. 4224
D. 5784

Answer

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

### Step 1: Understand the given information
- The perimeter (circumference) of the base of the cone is 132 cm.
- The height of the cone is 72 cm.

### Step 2: Relate the perimeter to the radius
The formula for the perimeter (circumference) of a circle is given by:
\[ C = 2 \pi r \]
Where \( r \) is the radius of the base of the cone.

Given that:
\[ 2 \pi r = 132 \]

### Step 3: Solve for the radius \( r \)
To find \( r \), we can rearrange the equation:
\[ r = \frac{132}{2 \pi} \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ r = \frac{132}{2 \times \frac{22}{7}} \]
\[ r = \frac{132 \times 7}{44} \]
\[ r = \frac{924}{44} \]
\[ r = 21 \text{ cm} \]

### Step 4: Find the slant height \( L \)
The slant height \( L \) of the cone can be calculated using the Pythagorean theorem:
\[ L = \sqrt{h^2 + r^2} \]
Where \( h \) is the height of the cone.

Substituting the values:
\[ L = \sqrt{72^2 + 21^2} \]
Calculating the squares:
\[ L = \sqrt{5184 + 441} \]
\[ L = \sqrt{5625} \]
\[ L = 75 \text{ cm} \]

### Step 5: Calculate the curved surface area (CSA)
The formula for the curved surface area of a cone is given by:
\[ \text{CSA} = \pi r L \]

Substituting the values:
\[ \text{CSA} = \frac{22}{7} \times 21 \times 75 \]
Calculating:
\[ \text{CSA} = \frac{22 \times 21 \times 75}{7} \]
\[ \text{CSA} = \frac{34650}{7} \]
\[ \text{CSA} = 4950 \text{ cm}^2 \]

### Final Answer
The curved surface area of the cone is \( 4950 \text{ cm}^2 \).

---