The radius of the base of a cone and also that of a hemisphere each is 14 cm . Also the volume of these two solids are equal. The curved surface area of the cone is
(Take ` pi (22)/( 7)` )

Options

A. `614 sqrt(5) cm^(2)`
B. `626 sqrt(5) cm^(2)`
C. `616 sqrt(5) cm^(2)`
D. `516 sqrt(5) cm^(2)`

Answer

To find the curved surface area of the cone given that the radius of both the cone and the hemisphere is 14 cm and their volumes are equal, we can follow these steps:

### Step 1: Write the formula for the volume of the cone and the hemisphere.
- The volume of a cone (V_cone) is given by:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
- The volume of a hemisphere (V_hemisphere) is given by:
\[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]

### Step 2: Set the volumes equal to each other.
Since the volumes of the cone and the hemisphere are equal:
\[ \frac{1}{3} \pi r^2 h = \frac{2}{3} \pi r^3 \]

### Step 3: Cancel out common terms.
We can cancel \(\frac{1}{3} \pi\) from both sides:
\[ r^2 h = 2 r^3 \]

### Step 4: Solve for height (h).
Dividing both sides by \(r^2\) (where \(r = 14\) cm):
\[ h = 2r \]
Substituting \(r = 14\) cm:
\[ h = 2 \times 14 = 28 \text{ cm} \]

### Step 5: Calculate the slant height (l) of the cone.
The slant height \(l\) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{14^2 + 28^2} = \sqrt{196 + 784} = \sqrt{980} \]
Calculating \(\sqrt{980}\):
\[ l = \sqrt{980} = 14\sqrt{5} \text{ cm} \approx 44.72 \text{ cm} \]

### Step 6: Calculate the curved surface area of the cone.
The curved surface area (CSA) of the cone is given by:
\[ CSA = \pi r l \]
Substituting the values:
\[ CSA = \frac{22}{7} \times 14 \times 14\sqrt{5} \]
Calculating:
\[ CSA = \frac{22}{7} \times 14 \times 14\sqrt{5} = \frac{22 \times 14^2 \sqrt{5}}{7} \]
Calculating \(14^2 = 196\):
\[ CSA = \frac{22 \times 196 \sqrt{5}}{7} \]
Calculating \(22 \times 196 = 4312\):
\[ CSA = \frac{4312 \sqrt{5}}{7} \approx 1376.16 \text{ cm}^2 \]

### Final Answer:
The curved surface area of the cone is approximately \(1376.16 \text{ cm}^2\).