The circumference of the base of a right circular cone is ` 220 cm` and its height 84 cm. The curved surface area of the cone is :

Options

A. `20020 cm^(2) `
B. `2020 cm^(2) `
C. `2200 cm^(2) `
D. `10010 cm^(2) `

Answer

To find the curved surface area of a right circular cone, we can follow these steps:

### Step 1: Find the radius of the base of the cone.
We know the circumference \( C \) of the base of the cone is given by the formula:
\[ C = 2 \pi r \]
Given that \( C = 220 \, \text{cm} \), we can set up the equation:
\[ 220 = 2 \pi r \]
To solve for \( r \), we can rearrange the equation:
\[ r = \frac{220}{2 \pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{220}{2 \times \frac{22}{7}} = \frac{220 \times 7}{44} = \frac{1540}{44} = 35 \, \text{cm} \]

### Step 2: Find the slant height of the cone.
The slant height \( l \) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
where \( h \) is the height of the cone. Given \( h = 84 \, \text{cm} \):
\[ l = \sqrt{35^2 + 84^2} \]
Calculating \( 35^2 \) and \( 84^2 \):
\[ 35^2 = 1225 \quad \text{and} \quad 84^2 = 7056 \]
Now, adding these values:
\[ l = \sqrt{1225 + 7056} = \sqrt{8281} \]
Calculating the square root:
\[ l = 91 \, \text{cm} \]

### Step 3: Calculate the curved surface area of the cone.
The formula for the curved surface area \( A \) of a cone is given by:
\[ A = \pi r l \]
Substituting the values of \( r \) and \( l \):
\[ A = \pi \times 35 \times 91 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 35 \times 91 \]
Calculating:
\[ A = \frac{22 \times 35 \times 91}{7} \]
Simplifying \( \frac{35}{7} = 5 \):
\[ A = 22 \times 5 \times 91 = 110 \times 91 = 10010 \, \text{cm}^2 \]

### Final Answer:
The curved surface area of the cone is \( 10010 \, \text{cm}^2 \).

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