Cylinder Hemisphere Radius 17A908

1. **State the problem:** We have a composite solid made of a cylinder joined to a hemisphere. The total surface area is given as $$176\pi$$ m². We need to find the radius $$r$$ of the cylinder.\n\n2. **Identify the surface areas involved:**\n- The hemisphere has surface area $$2\pi r^{2}$$ (since the surface area of a full sphere is $$4\pi r^{2}$$, half of that is $$2\pi r^{2}$$).\n- The cylinder has two circular ends and a curved surface area. However, one circular end is joined to the hemisphere, so it is not exposed.\n- The exposed surface area of the cylinder is the curved surface area plus the top circular base.\n- Curved surface area of cylinder = $$2\pi r h$$, where $$h$$ is the height of the cylinder.\n- Top circular base area = $$\pi r^{2}$$.\n\n3. **Given:**\n- Height of cylinder $$h = 4r$$.\n\n4. **Write the total surface area formula:**\n$$\text{Total surface area} = \text{surface area of hemisphere} + \text{curved surface area of cylinder} + \text{top base area}$$\n$$= 2\pi r^{2} + 2\pi r (4r) + \pi r^{2}$$\n\n5. **Simplify the expression:**\n$$= 2\pi r^{2} + 8\pi r^{2} + \pi r^{2} = (2 + 8 + 1)\pi r^{2} = 11\pi r^{2}$$\n\n6. **Set equal to given total surface area:**\n$$11\pi r^{2} = 176\pi$$\n\n7. **Divide both sides by $$\pi$$:**\n$$11 r^{2} = 176$$\n\n8. **Divide both sides by 11:**\n$$\cancel{11} r^{2} = \frac{176}{\cancel{11}}$$\n$$r^{2} = 16$$\n\n9. **Take the square root of both sides:**\n$$r = \sqrt{16} = 4$$\n\n**Final answer:** The radius $$r$$ of the cylinder is $$4$$ meters.