The volume of a right circular cone is 2464 `cm^2` If the radius of its base is 14 cm, then its curved surface area (in `cm^2` ) is: (Take`pi =(22)/(7)`)

Options

A. `88 sqrt(58)`
B. `44 sqrt(58)`
C. `44 sqrt(85)`
D. `88 sqrt(85)`

Answer

To find the curved surface area of the right circular cone given its volume and radius, we can follow these steps:

### Step 1: Use the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cone.

### Step 2: Substitute the known values into the volume formula
Given:
- Volume \( V = 2464 \, \text{cm}^3 \)
- Radius \( r = 14 \, \text{cm} \)
- \( \pi = \frac{22}{7} \)

Substituting these values into the volume formula:
\[ 2464 = \frac{1}{3} \times \frac{22}{7} \times (14)^2 \times h \]

### Step 3: Simplify the equation
First, calculate \( (14)^2 \):
\[ (14)^2 = 196 \]
Now substituting this back into the equation:
\[ 2464 = \frac{1}{3} \times \frac{22}{7} \times 196 \times h \]

### Step 4: Calculate \( \frac{22}{7} \times 196 \)
Calculating \( \frac{22 \times 196}{7} \):
\[ \frac{22 \times 196}{7} = \frac{4312}{7} = 616 \]
Now the equation becomes:
\[ 2464 = \frac{1}{3} \times 616 \times h \]

### Step 5: Multiply both sides by 3
To eliminate the fraction, multiply both sides by 3:
\[ 7392 = 616h \]

### Step 6: Solve for \( h \)
Now, divide both sides by 616:
\[ h = \frac{7392}{616} = 12 \, \text{cm} \]

### Step 7: 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 of \( r \) and \( h \):
\[ l = \sqrt{14^2 + 12^2} = \sqrt{196 + 144} = \sqrt{340} \]

### Step 8: Calculate the curved surface area (CSA)
The formula for the curved surface area (CSA) of a cone is:
\[ \text{CSA} = \pi r l \]
Substituting the values of \( \pi \), \( r \), and \( l \):
\[ \text{CSA} = \frac{22}{7} \times 14 \times \sqrt{340} \]

### Step 9: Simplify the CSA expression
Calculating \( \frac{22 \times 14}{7} \):
\[ \frac{22 \times 14}{7} = 44 \]
Thus, the CSA becomes:
\[ \text{CSA} = 44 \sqrt{340} \]

### Step 10: Final Calculation
To simplify \( \sqrt{340} \):
\[ \sqrt{340} = \sqrt{4 \times 85} = 2\sqrt{85} \]
Thus, the CSA is:
\[ \text{CSA} = 44 \times 2\sqrt{85} = 88\sqrt{85} \, \text{cm}^2 \]

### Final Answer
The curved surface area of the cone is:
\[ \text{CSA} = 88\sqrt{85} \, \text{cm}^2 \]