1. **State the problem:** We have a right circular cylinder full of ice cream with diameter 12 cm and height 15 cm. We want to find how many cones (each with a hemispherical top) of height 12 cm and diameter 6 cm can be filled with the ice cream from the cylinder. 2. **Formulas and important rules:** - Volume of cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Volume of hemisphere: $$V_{hemisphere} = \frac{2}{3} \pi r^3$$ 3. **Calculate the volume of the cylinder:** - Diameter = 12 cm, so radius $$r = \frac{12}{2} = 6$$ cm - Height $$h = 15$$ cm - Volume $$V_{cyl} = \pi \times 6^2 \times 15 = \pi \times 36 \times 15 = 540\pi$$ cm³ 4. **Calculate the volume of one cone with hemispherical top:** - Diameter = 6 cm, so radius $$r = \frac{6}{2} = 3$$ cm - Height of cone part $$h = 12$$ cm - Volume of cone part: $$V_{cone} = \frac{1}{3} \pi \times 3^2 \times 12 = \frac{1}{3} \pi \times 9 \times 12 = 36\pi$$ cm³ - Volume of hemisphere part: $$V_{hemisphere} = \frac{2}{3} \pi \times 3^3 = \frac{2}{3} \pi \times 27 = 18\pi$$ cm³ - Total volume of one cone with hemisphere: $$V_{total} = 36\pi + 18\pi = 54\pi$$ cm³ 5. **Find the number of such cones that can be filled:** - Number of cones $$= \frac{V_{cyl}}{V_{total}} = \frac{540\pi}{54\pi} = 10$$ **Final answer:** $$\boxed{10}$$ cones can be filled with the ice cream from the cylinder.