1. **Problem statement:** A solid toy is made by mounting a cone on top of a hemisphere. The height of the cone is 7 cm and the radius of both the cone and hemisphere is 3.5 cm. We need to find the total surface area (TSA) and volume of the solid. 2. **Formulas and important rules:** - Volume of cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Volume of hemisphere: $$V_{hemisphere} = \frac{2}{3} \pi r^3$$ - Total volume of solid: $$V = V_{cone} + V_{hemisphere}$$ - Curved surface area of cone: $$A_{cone} = \pi r l$$ where $$l = \sqrt{r^2 + h^2}$$ is the slant height. - Surface area of hemisphere: $$A_{hemisphere} = 2 \pi r^2$$ - Total surface area (TSA) of solid = curved surface area of cone + surface area of hemisphere (base of cone is not exposed as it is mounted on hemisphere) 3. **Calculate slant height of cone:** $$l = \sqrt{3.5^2 + 7^2} = \sqrt{12.25 + 49} = \sqrt{61.25} \approx 7.83$$ cm 4. **Calculate curved surface area of cone:** $$A_{cone} = \pi \times 3.5 \times 7.83 \approx 3.1416 \times 3.5 \times 7.83 = 86.01$$ cm\textsuperscript{2} 5. **Calculate surface area of hemisphere:** $$A_{hemisphere} = 2 \pi \times 3.5^2 = 2 \times 3.1416 \times 12.25 = 76.97$$ cm\textsuperscript{2} 6. **Calculate total surface area (TSA):** $$TSA = A_{cone} + A_{hemisphere} = 86.01 + 76.97 = 162.98$$ cm\textsuperscript{2} 7. **Calculate volume of cone:** $$V_{cone} = \frac{1}{3} \pi \times 3.5^2 \times 7 = \frac{1}{3} \times 3.1416 \times 12.25 \times 7 = \frac{1}{3} \times 3.1416 \times 85.75 = 89.79$$ cm\textsuperscript{3} 8. **Calculate volume of hemisphere:** $$V_{hemisphere} = \frac{2}{3} \pi \times 3.5^3 = \frac{2}{3} \times 3.1416 \times 42.875 = 89.79$$ cm\textsuperscript{3} 9. **Calculate total volume:** $$V = V_{cone} + V_{hemisphere} = 89.79 + 89.79 = 179.58$$ cm\textsuperscript{3} **Final answers:** - Total surface area = 162.98 cm\textsuperscript{2} - Total volume = 179.58 cm\textsuperscript{3}