The internal and external diameters of a hollow hemispherical bowl are 18 cm and 22 cm, respectively. The total surface area (in `cm^(2)`) of the bowl is:

Options

A. 424 `pi`
B. 414 `pi`
C. 444 `pi`
D. 404 `pi`

Answer

To find the total surface area of a hollow hemispherical bowl, we need to consider both the internal and external surfaces of the bowl. The formula for the total surface area of a hollow hemispherical bowl includes the curved surface area of both the inner and outer hemispheres, as well as the area of the circular base.

### Step-by-step Solution:

1. **Identify the Diameters**:
- Internal diameter (d1) = 18 cm
- External diameter (d2) = 22 cm

2. **Calculate the Radii**:
- Internal radius (r1) = d1 / 2 = 18 cm / 2 = 9 cm
- External radius (r2) = d2 / 2 = 22 cm / 2 = 11 cm

3. **Calculate the Curved Surface Area**:
- The curved surface area (CSA) of a hemisphere is given by the formula:
\[ \text{CSA} = 2\pi r^2 \]
- For the internal hemisphere:
\[ \text{CSA}_{\text{internal}} = 2\pi (9^2) = 2\pi (81) = 162\pi \, \text{cm}^2 \]
- For the external hemisphere:
\[ \text{CSA}_{\text{external}} = 2\pi (11^2) = 2\pi (121) = 242\pi \, \text{cm}^2 \]

4. **Calculate the Area of the Circular Base**:
- The area of the circular base is given by the formula:
\[ \text{Area} = \pi r^2 \]
- For the internal base:
\[ \text{Area}_{\text{internal}} = \pi (9^2) = \pi (81) = 81\pi \, \text{cm}^2 \]
- For the external base:
\[ \text{Area}_{\text{external}} = \pi (11^2) = \pi (121) = 121\pi \, \text{cm}^2 \]

5. **Calculate the Total Surface Area**:
- The total surface area (TSA) of the hollow hemispherical bowl is the sum of the curved surface areas and the areas of the bases:
\[ \text{TSA} = \text{CSA}_{\text{internal}} + \text{CSA}_{\text{external}} + \text{Area}_{\text{external}} - \text{Area}_{\text{internal}} \]
- Substituting the values:
\[ \text{TSA} = 162\pi + 242\pi + 121\pi - 81\pi \]
\[ \text{TSA} = (162 + 242 + 121 - 81)\pi = 444\pi \, \text{cm}^2 \]

6. **Final Calculation**:
- Approximating \(\pi\) as 3.14:
\[ \text{TSA} \approx 444 \times 3.14 \approx 1397.76 \, \text{cm}^2 \]

### Conclusion:
The total surface area of the hollow hemispherical bowl is approximately **1397.76 cm²**.