Identify the dimensions and components of the surface area in contact with water.

The container has an internal radius $r = 10.5$ cm.
The water height is $H = 30.5$ cm.
The base is hemispherical, so its height is equal to its radius, $r = 10.5$ cm.
The height of the cylindrical part filled with water is $h = H - r = 30.5 \text{ cm} - 10.5 \text{ cm} = 20 \text{ cm}$.
The surface area in contact with water consists of three parts:

1. The curved surface area of the hemispherical base.
2. The curved surface area of the cylindrical part up to the water level.
3. The area of the water surface (a circular area at the top of the water).

The formula for the curved surface area of a hemisphere is $2\pi r^2$.

$$A_{hemisphere} = 2 \times \frac{22}{7} \times (10.5 \text{ cm})^2$$ $$A_{hemisphere} = 2 \times \frac{22}{7} \times 110.25 \text{ cm}^2$$ $$A_{hemisphere} = 44 \times \frac{110.25}{7} \text{ cm}^2$$ $$A_{hemisphere} = 44 \times 15.75 \text{ cm}^2$$ $$A_{hemisphere} = 693 \text{ cm}^2$$

The formula for the curved surface area of a cylinder is $2\pi r h$.

$$A_{cylinder} = 2 \times \frac{22}{7} \times 10.5 \text{ cm} \times 20 \text{ cm}$$ $$A_{cylinder} = 2 \times \frac{22}{7} \times \frac{21}{2} \text{ cm} \times 20 \text{ cm}$$ $$A_{cylinder} = 22 \times 3 \times 20 \text{ cm}^2$$ $$A_{cylinder} = 66 \times 20 \text{ cm}^2$$ $$A_{cylinder} = 1320 \text{ cm}^2$$

The formula for the area of a circle is $\pi r^2$.

$$A_{water\_surface} = \frac{22}{7} \times (10.5 \text{ cm})^2$$ $$A_{water\_surface} = \frac{22}{7} \times 110.25 \text{ cm}^2$$ $$A_{water\_surface} = 22 \times 15.75 \text{ cm}^2$$ $$A_{water\_surface} = 346.5 \text{ cm}^2$$

The total surface area is the sum of the areas calculated in Steps 2, 3, and 4.

$$A_{total} = A_{hemisphere} + A_{cylinder} + A_{water\_surface}$$ $$A_{total} = 693 \text{ cm}^2 + 1320 \text{ cm}^2 + 346.5 \text{ cm}^2$$ $$A_{total} = 2359.5 \text{ cm}^2$$

The surface area of the container that is in contact with water is $\boxed{\text{2359.5 cm}^2}$.