Surface Area Cylinder Hemisphere 2F7837

1. **Problem Statement:** Find the surface area of the composite solid consisting of a cylinder with a hemisphere on top. Given: - Diameter of the base (and hemisphere) = $12$ in. - Height of the cylinder = $13$ in. 2. **Formulas and Important Rules:** - Radius $r = \frac{12}{2} = 6$ in. - Surface area of cylinder (excluding top base): $$2\pi r h$$ - Surface area of hemisphere (curved surface only): $$2\pi r^2$$ - The top circular base of the cylinder is covered by the hemisphere, so it is not counted separately. 3. **Calculate each part:** - Cylinder lateral surface area: $$2\pi \times 6 \times 13 = 156\pi$$ - Hemisphere curved surface area: $$2\pi \times 6^2 = 2\pi \times 36 = 72\pi$$ 4. **Total surface area:** Since the base of the cylinder is exposed, add the area of the bottom circle: - Base area: $$\pi r^2 = \pi \times 6^2 = 36\pi$$ So total surface area is: $$156\pi + 72\pi + 36\pi = 264\pi$$ 5. **Final answer:** $$\text{Surface Area} = 264\pi \approx 264 \times 3.1416 = 829.38 \text{ square inches}$$ This includes: - 1 circular base - 1 curved lateral surface of the cylinder - 1 curved surface of the hemisphere Hence, the composite solid has 3 faces: 1 circular face, 2 curved faces.