The area of the base of a right circular cone is `(1408)/(7) cm^(2)` and its height is 6 cm. Taking `pi = (22)/(7)`, what will be the curved surface area of the cone ?

Options

A. `(1760)/(7) cm^(2)`
B. `(1067)/(7) cm^(2)`
C. `(1670)/(7) cm^(2)`
D. `(7160)/(7) cm^(2)`

Answer

To find the curved surface area (CSA) of the right circular cone, we will follow these steps:

### Step 1: Understand the given data
We are given:
- Area of the base of the cone = \( \frac{1408}{7} \, \text{cm}^2 \)
- Height of the cone (h) = 6 cm
- Value of \( \pi = \frac{22}{7} \)

### Step 2: Use the area of the base to find the radius
The area of the base of a cone is given by the formula:
\[ \text{Area} = \pi r^2 \]
Substituting the values we have:
\[ \frac{1408}{7} = \frac{22}{7} r^2 \]
To eliminate \( \frac{7}{7} \), we can multiply both sides by 7:
\[ 1408 = 22 r^2 \]
Now, divide both sides by 22 to solve for \( r^2 \):
\[ r^2 = \frac{1408}{22} \]
Calculating \( \frac{1408}{22} \):
\[ r^2 = 64 \]
Taking the square root to find \( r \):
\[ r = \sqrt{64} = 8 \, \text{cm} \]

### Step 3: Calculate the slant height (l)
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values of \( h \) and \( r \):
\[ l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{cm} \]

### Step 4: 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 of \( \pi \), \( r \), and \( l \):
\[ \text{CSA} = \frac{22}{7} \times 8 \times 10 \]
Calculating this:
\[ \text{CSA} = \frac{22 \times 80}{7} = \frac{1760}{7} \, \text{cm}^2 \]

### Final Answer
The curved surface area of the cone is:
\[ \text{CSA} = \frac{1760}{7} \, \text{cm}^2 \]

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