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 surface area ( in `cm^(2)`) of the cone ?

Options

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

Answer

To find the 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
We know that:
- The perimeter (circumference) of the base of the cone is 132 cm.
- The height (h) of the cone is 72 cm.

### Step 2: Find the Radius of the Base
The formula for the circumference (C) of a circle is given by:
\[ C = 2\pi r \]
Where \( r \) is the radius.

Given that the circumference is 132 cm, we can set up the equation:
\[ 2\pi r = 132 \]

Using \( \pi \approx \frac{22}{7} \), we can substitute:
\[ 2 \times \frac{22}{7} \times r = 132 \]

Now, simplify this equation:
\[ \frac{44}{7} r = 132 \]

To isolate \( r \), multiply both sides by \( \frac{7}{44} \):
\[ r = 132 \times \frac{7}{44} \]

Calculating this gives:
\[ r = 132 \div 44 \times 7 = 3 \times 7 = 21 \text{ cm} \]

### Step 3: Calculate the Slant Height (l)
The slant height \( l \) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]

Substituting the values of \( h \) and \( r \):
\[ l = \sqrt{72^2 + 21^2} \]
\[ l = \sqrt{5184 + 441} \]
\[ l = \sqrt{5625} \]
\[ l = 75 \text{ cm} \]

### Step 4: Calculate the Surface Area
The total surface area (A) of a cone is given by the formula:
\[ A = \pi r (l + r) \]

Substituting the values we have:
\[ A = \frac{22}{7} \times 21 \times (75 + 21) \]
\[ A = \frac{22}{7} \times 21 \times 96 \]

Now, calculate:
\[ A = \frac{22 \times 21 \times 96}{7} \]

First, calculate \( 22 \times 21 = 462 \):
\[ A = \frac{462 \times 96}{7} \]

Now, calculate \( 462 \times 96 = 44352 \):
\[ A = \frac{44352}{7} = 6336 \text{ cm}^2 \]

### Final Answer
The surface area of the cone is \( 6336 \text{ cm}^2 \).

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