1. **Problem statement:** We have a cone with apex A and a cylinder inscribed inside it. Given AG = 12 cm (height of the cone), AC = 2\sqrt{3} cm (radius of the cylinder's top circle), and \angle AGE = 60^\circ. We need to find the height of the cylinder. 2. **Understanding the problem:** The cylinder is inside the cone, so its radius CD is less than the cone's base radius FG. The height of the cylinder is what we want to find. 3. **Key observations:** - AG is the slant height of the cone. - AC is the radius of the cylinder's top circle. - \angle AGE = 60^\circ is the angle between AG and GE. 4. **Using trigonometry:** - In triangle AGE, AG = 12 cm, \angle AGE = 60^\circ. - We can find GE using the cosine rule or by considering the right triangle if applicable. 5. **Calculate GE:** Since \angle AGE = 60^\circ and AG = 12 cm, $$ GE = AG \times \cos 60^\circ = 12 \times \frac{1}{2} = 6 \text{ cm} $$ 6. **Calculate AE:** $$ AE = AG \times \sin 60^\circ = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \text{ cm} $$ 7. **Relate AC and AE:** AC = 2\sqrt{3} cm is the radius of the cylinder's top circle, which lies on the segment AE. 8. **Find the height of the cylinder:** The height of the cylinder is the vertical distance from the base to the top circle inside the cone. Since AE is the vertical height from A to the base, and AC is the radius at that height, the height of the cylinder is $$ h = AE - AC = 6\sqrt{3} - 2\sqrt{3} = 4\sqrt{3} \text{ cm} $$ **Final answer:** The height of the cylinder is $4\sqrt{3}$ cm, which corresponds to option D.