1. **State the problem:** Calculate the total surface area of the chocolate painted object, which consists of a closed cylinder with an open hemisphere inside its top. 2. **Given data:** - Radius of cylinder and hemisphere: $r = 36$ mm - Height of cylinder (excluding hemisphere): $h = 68$ mm 3. **Formulas used:** - Surface area of closed cylinder (excluding top): $$A_{cyl} = 2\pi r h + \pi r^2$$ where $2\pi r h$ is the lateral surface area and $\pi r^2$ is the bottom area. - Surface area of hemisphere (outer curved surface only): $$A_{hem} = 2\pi r^2$$ 4. **Important note:** The hemisphere is inside the top of the cylinder, so the top circular area of the cylinder is not visible and is replaced by the inner hemisphere surface. The painted surface includes: - The lateral surface of the cylinder - The bottom of the cylinder - The outer curved surface of the hemisphere 5. **Calculate each area:** - Cylinder lateral surface area: $$2\pi r h = 2 \times \pi \times 36 \times 68 = 4896\pi$$ - Cylinder bottom area: $$\pi r^2 = \pi \times 36^2 = 1296\pi$$ - Hemisphere curved surface area: $$2\pi r^2 = 2 \times \pi \times 36^2 = 2592\pi$$ 6. **Sum all painted areas:** $$A_{total} = 4896\pi + 1296\pi + 2592\pi = (4896 + 1296 + 2592)\pi = 8784\pi$$ 7. **Calculate numerical value:** $$A_{total} \approx 8784 \times 3.1416 = 27588.5 \text{ mm}^2$$ **Final answer:** The total painted surface area is approximately **27588.5 mm²**.