The area of the base of a right circular cone is `154` `cm^2` and its height is `14 cm` . Its curved surface area is

Options

A. `154sqrt5 cm^2`
B. `154 sqrt7 cm^2`
C. `77 sqrt7 cm^2`
D. `77 sqrt5 cm^2`

Explanation

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

### Step 1: Identify the given values
We know the following:
- Area of the base of the cone (A) = 154 cm²
- Height of the cone (h) = 14 cm

### Step 2: Relate the base area to the radius
The area of the base of a cone is given by the formula:
\[ A = \pi r^2 \]
Where \( r \) is the radius of the base. We can rearrange this to find \( r \):
\[ r^2 = \frac{A}{\pi} \]

### Step 3: Substitute the values
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{154}{\frac{22}{7}} \]
\[ r^2 = 154 \times \frac{7}{22} \]
\[ r^2 = 49 \]
Taking the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 4: Calculate the slant height (l)
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{7^2 + 14^2} \]
\[ l = \sqrt{49 + 196} \]
\[ l = \sqrt{245} \]
\[ l = 7\sqrt{5} \, \text{cm} \]

### Step 5: Calculate the curved surface area (CSA)
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi r l \]
Substituting the values we have:
\[ \text{CSA} = \frac{22}{7} \times 7 \times 7\sqrt{5} \]
\[ \text{CSA} = 22 \times 7\sqrt{5} \]
\[ \text{CSA} = 154\sqrt{5} \, \text{cm}^2 \]

### Final Answer
The curved surface area of the cone is \( 154\sqrt{5} \, \text{cm}^2 \).

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