To solve this problem, we need to calculate the total surface area of the combined shape, which includes a cylindrical base and a hemispherical dome on top. Here's the step-by-step solution:

Given Data

• Radius of both the hemisphere and the cylinder, $r = 20$ cm
• Height of the cylinder, $h = 50$ cm

Step 1: Surface Area of the Cylinder

The surface area of the cylinder consists of:

1. The lateral (curved) surface area.
2. The bottom circular base area.

The formula for the lateral surface area of a cylinder is: $\text{Lateral Surface Area (Cylinder)} = 2 \pi r h$ Substitute $r = 20$ and $h = 50$: $\text{Lateral Surface Area} = 2 \pi \times 20 \times 50 = 2000 \pi \, \text{cm}^2$

The area of the circular base (bottom) of the cylinder is: $\text{Base Area} = \pi r^2$ $\text{Base Area} = \pi \times 20^2 = 400 \pi \, \text{cm}^2$

Step 2: Surface Area of the Hemisphere

The hemisphere has only its curved surface (since the flat side is attached to the cylinder's top and doesn't count as an external surface). The formula for the curved surface area of a hemisphere is: $\text{Curved Surface Area (Hemisphere)} = 2 \pi r^2$ Substitute $r = 20$: $\text{Curved Surface Area} = 2 \pi \times 20^2 = 800 \pi \, \text{cm}^2$

Step 3: Total Surface Area

Add up the areas from the cylinder's lateral surface, the cylinder's base, and the hemisphere's curved surface: $\text{Total Surface Area} = 2000 \pi + 400 \pi + 800 \pi = 3200 \pi \, \text{cm}^2$

Step 4: Calculating the Approximate Value

Using $\pi \approx 3.14159$: $\text{Total Surface Area} \approx 3200 \times 3.14159 = 10053.096 \, \text{cm}^2$ Rounded to the nearest square centimeter, the total surface area is: $\boxed{10053 \, \text{cm}^2}$

Would you like further details, or do you have any questions?

Here are some related questions to expand on this topic:

1. How would the surface area change if the height of the cylinder increased?
2. What if the radius of the hemisphere was different from the cylinder's radius?
3. How do we calculate the volume of this combined shape?
4. Can we use a similar approach to find the surface area of a cone-capped cylinder?
5. How would this calculation change if the shape had an open top?

Tip: Always double-check the boundaries of shapes in composite solids to ensure only the exposed surfaces are included in surface area calculations.