The capacity and the base area of a right circular conical vessel are 9856 `cm^(3)` and 616 `cm^(2)` respectively. Find the curved surface area of the vessel.

Answer

To find the curved surface area of a right circular conical vessel given its capacity and base area, we can follow these steps:

### Step 1: Understand the Given Information
- Capacity (Volume) of the conical vessel, \( V = 9856 \, \text{cm}^3 \)
- Base area of the conical vessel, \( A = 616 \, \text{cm}^2 \)

### Step 2: Use the Formula for 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
- \( h \) is the height of the cone

### Step 3: Relate Base Area to Radius
The base area \( A \) of the cone can be expressed as:
\[ A = \pi r^2 \]
Given that \( A = 616 \, \text{cm}^2 \), we can set up the equation:
\[ \pi r^2 = 616 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 616 \]

### Step 4: Solve for \( r^2 \)
Multiply both sides by \( 7 \):
\[ 22 r^2 = 616 \times 7 \]
Calculating \( 616 \times 7 \):
\[ 616 \times 7 = 4312 \]
So,
\[ 22 r^2 = 4312 \]
Now, divide by \( 22 \):
\[ r^2 = \frac{4312}{22} = 196 \]

### Step 5: Find the Radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Step 6: Use Volume to Find Height \( h \)
Now we can substitute \( r \) back into the volume formula to find \( h \):
\[ 9856 = \frac{1}{3} \pi (14^2) h \]
Calculating \( 14^2 \):
\[ 14^2 = 196 \]
Thus,
\[ 9856 = \frac{1}{3} \pi (196) h \]
Substituting \( \pi \approx \frac{22}{7} \):
\[ 9856 = \frac{1}{3} \times \frac{22}{7} \times 196 \times h \]
Calculating \( \frac{22 \times 196}{3 \times 7} \):
\[ \frac{22 \times 196}{21} = \frac{4312}{21} \]
So we have:
\[ 9856 = \frac{4312}{21} h \]
Now, solve for \( h \):
\[ h = \frac{9856 \times 21}{4312} \]
Calculating \( \frac{9856 \times 21}{4312} \):
\[ h = 14 \, \text{cm} \]

### Step 7: Calculate the Curved Surface Area
The curved surface area \( CSA \) of a cone is given by:
\[ CSA = \pi r l \]
Where \( l \) is the slant height, which can be found using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Calculating \( l \):
\[ l = \sqrt{14^2 + 14^2} = \sqrt{196 + 196} = \sqrt{392} = 14\sqrt{2} \]

### Step 8: Substitute \( r \) and \( l \) into the CSA Formula
Now substituting \( r \) and \( l \) into the CSA formula:
\[ CSA = \pi \times 14 \times 14\sqrt{2} \]
Using \( \pi \approx \frac{22}{7} \):
\[ CSA = \frac{22}{7} \times 14 \times 14\sqrt{2} \]
Calculating:
\[ CSA = 22 \times 28\sqrt{2} = 616\sqrt{2} \, \text{cm}^2 \]

### Final Answer
The curved surface area of the conical vessel is:
\[ CSA \approx 616 \times 1.414 \approx 871.68 \, \text{cm}^2 \]