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)`, the curved surface area of the cone is:

Options

A. `(1670 )/(7) cm^2`
B. `(1067 )/(7) cm^2`
C. `(1760)/(7) cm^2`
D. ` (7160 )/(7) 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
The area of the base of the cone is given by the formula:
\[ \text{Area of base} = \pi r^2 \]
We know that the area of the base is \(\frac{1408}{7} \, \text{cm}^2\) and \(\pi = \frac{22}{7}\). Therefore, we can set up the equation:
\[ \frac{22}{7} r^2 = \frac{1408}{7} \]

### Step 2: Solve for \(r^2\)
To eliminate the fraction, we can multiply both sides by 7:
\[ 22 r^2 = 1408 \]
Now, divide both sides by 22:
\[ r^2 = \frac{1408}{22} \]
Calculating this gives:
\[ r^2 = 64 \]

### Step 3: Find the radius \(r\)
Taking the square root of both sides, we find:
\[ r = \sqrt{64} = 8 \, \text{cm} \]

### Step 4: Find the slant height \(l\)
The slant height \(l\) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
where \(h\) is the height of the cone (6 cm). Plugging in the values:
\[ l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{cm} \]

### Step 5: Calculate the curved surface area
The formula for the curved surface area (CSA) of a cone is:
\[ \text{CSA} = \pi r l \]
Substituting the known values:
\[ \text{CSA} = \frac{22}{7} \times 8 \times 10 \]
Calculating this gives:
\[ \text{CSA} = \frac{1760}{7} \, \text{cm}^2 \]

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

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