1. **Problem statement:** We have a solid right circular cylinder with height $12$ cm and base radius $5$ cm. A right circular cone with the same height and base radius is removed from it. We need to find the volume and total surface area of the remaining solid. 2. **Formulas used:** - Volume of cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Surface area of cylinder (including top and bottom): $$A_{cyl} = 2\pi r h + 2\pi r^2$$ - Surface area of cone (lateral area only, since base is removed): $$A_{cone} = \pi r l$$ where $l$ is the slant height. 3. **Calculate volumes:** - Cylinder volume: $$V_{cyl} = 3.14 \times 5^2 \times 12 = 3.14 \times 25 \times 12 = 942$$ cm$^3$ - Cone volume: $$V_{cone} = \frac{1}{3} \times 3.14 \times 25 \times 12 = \frac{1}{3} \times 942 = 314$$ cm$^3$ - Remaining volume: $$V_{rem} = V_{cyl} - V_{cone} = 942 - 314 = 628$$ cm$^3$ 4. **Calculate surface areas:** - Cylinder surface area: $$A_{cyl} = 2 \times 3.14 \times 5 \times 12 + 2 \times 3.14 \times 5^2 = 2 \times 3.14 \times 5 \times 12 + 2 \times 3.14 \times 25 = 376.8 + 157 = 533.8$$ cm$^2$ - Slant height of cone: $$l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$ cm - Cone lateral surface area: $$A_{cone} = 3.14 \times 5 \times 13 = 204.1$$ cm$^2$ 5. **Remaining solid surface area:** - The base of the cone is removed from the cylinder's top, so the top circular area is replaced by the cone's lateral surface. - Total surface area of remaining solid: $$A_{rem} = \text{cylinder lateral area} + \text{bottom area} + \text{cone lateral area}$$ - Cylinder lateral area: $$2 \pi r h = 2 \times 3.14 \times 5 \times 12 = 376.8$$ cm$^2$ - Bottom area: $$\pi r^2 = 3.14 \times 25 = 78.5$$ cm$^2$ - Add cone lateral area: $$204.1$$ cm$^2$ - Sum: $$A_{rem} = 376.8 + 78.5 + 204.1 = 659.4$$ cm$^2$ **Final answers:** - Volume of remaining solid: $628$ cm$^3$ - Total surface area of remaining solid: $659.4$ cm$^2$