1. **Problem statement:** Find the volume of a solid consisting of a cone with a half-sphere on top. The radius of both the cone and the half-sphere is 10 cm, and the height of the cone is 20 cm. 2. **Formulas used:** - Volume of a cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Volume of a sphere: $$V_{sphere} = \frac{4}{3} \pi r^3$$ - Volume of a half-sphere: $$V_{half-sphere} = \frac{1}{2} V_{sphere} = \frac{2}{3} \pi r^3$$ 3. **Calculate the volume of the cone:** $$V_{cone} = \frac{1}{3} \pi (10)^2 (20) = \frac{1}{3} \pi \times 100 \times 20 = \frac{2000}{3} \pi$$ 4. **Calculate the volume of the half-sphere:** $$V_{half-sphere} = \frac{2}{3} \pi (10)^3 = \frac{2}{3} \pi \times 1000 = \frac{2000}{3} \pi$$ 5. **Total volume:** $$V_{total} = V_{cone} + V_{half-sphere} = \frac{2000}{3} \pi + \frac{2000}{3} \pi = \frac{4000}{3} \pi$$ 6. **Final answer:** $$V_{total} = \frac{4000}{3} \pi \approx 4188.79 \text{ cm}^3$$ The total volume of the solid is approximately 4188.79 cubic centimeters.