A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

Answer

To find the inner surface area of the vessel, which consists of a hollow hemisphere mounted on a hollow cylinder, we can follow these steps:

### Step 1: Identify the dimensions
- The diameter of the hemisphere is given as 14 cm.
- Therefore, the radius \( r \) of the hemisphere (and the cylinder) is:
\[ r = \frac{14}{2} = 7 \text{ cm} \]
- The total height of the vessel is given as 13 cm. Since the radius of the hemisphere is 7 cm, the height of the cylinder \( h \) can be calculated as:
\[ h = \text{Total height} - \text{Radius of hemisphere} = 13 - 7 = 6 \text{ cm} \]

### Step 2: Calculate the inner surface area
The inner surface area of the vessel consists of the curved surface area of the cylinder and the curved surface area of the hemisphere.

1. **Curved Surface Area of the Cylinder**:
The formula for the curved surface area (CSA) of a cylinder is:
\[ \text{CSA}_{\text{cylinder}} = 2\pi rh \]
Substituting the values:
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 7 \times 6 \]

2. **Curved Surface Area of the Hemisphere**:
The formula for the curved surface area of a hemisphere is:
\[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \]
Substituting the values:
\[ \text{CSA}_{\text{hemisphere}} = 2 \times \frac{22}{7} \times 7^2 \]

### Step 3: Combine the areas
Now, we can combine both surface areas:
\[ \text{Total Inner Surface Area} = \text{CSA}_{\text{cylinder}} + \text{CSA}_{\text{hemisphere}} \]

### Step 4: Simplify the calculations
1. Calculate the CSA of the cylinder:
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 7 \times 6 = 2 \times 22 \times 6 = 264 \text{ cm}^2 \]

2. Calculate the CSA of the hemisphere:
\[ \text{CSA}_{\text{hemisphere}} = 2 \times \frac{22}{7} \times 7^2 = 2 \times \frac{22}{7} \times 49 = 2 \times 22 \times 7 = 308 \text{ cm}^2 \]

3. Now combine both areas:
\[ \text{Total Inner Surface Area} = 264 + 308 = 572 \text{ cm}^2 \]

### Final Answer
The inner surface area of the vessel is:
\[ \boxed{572 \text{ cm}^2} \]