1. **Stating the problem:** We have a cone with vertex $A$ and base radius $FG$. Inside the cone is a cylinder with top circle radius $CD$. Given $AG=12$ cm, $AC=2\sqrt{3}$ cm, and $\angle AGE=60^\circ$, we need to find the length $FG$. 2. **Understanding the geometry:** Points $A$, $C$, and $G$ form a triangle with $\angle AGE=60^\circ$. $AG$ is a segment from vertex $A$ to point $G$ on the base circle of the cone. $AC$ is a segment from $A$ to $C$ on the cylinder's top circle. 3. **Using the Law of Cosines in triangle $AGE$:** Since $\angle AGE=60^\circ$, and $AG=12$, $AC=2\sqrt{3}$, we can find $GE$ or relate the sides to find $FG$. 4. **Relating the radii:** $FG$ is the radius of the cone's base circle, and $CD$ is the radius of the cylinder's top circle. Since the cylinder is inside the cone, the radius $CD$ is smaller than $FG$. 5. **Using the right triangle formed by $A$, $C$, and $G$:** We can use the Pythagorean theorem or trigonometric relations to find $FG$. 6. **Calculate $FG$:** Since $AC=2\sqrt{3}$ and $AG=12$, and $\angle AGE=60^\circ$, we use the Law of Cosines in triangle $AGE$: $$GE^2 = AG^2 + AE^2 - 2 \cdot AG \cdot AE \cdot \cos(60^\circ)$$ But $AE$ is unknown, so instead, consider the vertical height and radius relations. 7. **Using the cone's height and radius relation:** Assuming $A$ is the apex and $FG$ is the base radius, the height $h$ can be found from $AG$ and $AC$. 8. **Calculate $FG$ using similarity:** The triangle formed by $A$, $C$, and $G$ is similar to the triangle formed by $A$, $F$, and $G$ (the cone's base). Using similarity ratios: $$\frac{AC}{AG} = \frac{CD}{FG}$$ Given $AC=2\sqrt{3}$, $AG=12$, and $CD$ is the radius of the cylinder's top circle (equal to $AC$?), we solve for $FG$: $$FG = \frac{AG \cdot CD}{AC}$$ But $CD$ is unknown; however, since $CD$ is the radius of the cylinder's top circle and $AC$ is the length from $A$ to $C$, we can infer $CD=AC$. 9. **Final calculation:** $$FG = \frac{12 \cdot 2\sqrt{3}}{2\sqrt{3}} = 12$$ This contradicts the options, so re-examine. 10. **Alternative approach:** Using the triangle $AGE$ with $\angle AGE=60^\circ$, $AG=12$, and $AC=2\sqrt{3}$, find $GE$ using Law of Cosines: $$GE^2 = AG^2 + AC^2 - 2 \cdot AG \cdot AC \cdot \cos(60^\circ)$$ $$GE^2 = 12^2 + (2\sqrt{3})^2 - 2 \cdot 12 \cdot 2\sqrt{3} \cdot \frac{1}{2}$$ $$GE^2 = 144 + 12 - 24\sqrt{3}$$ 11. **Calculate $GE^2$ numerically:** $$GE^2 = 156 - 24\sqrt{3}$$ Approximate $\sqrt{3} \approx 1.732$: $$GE^2 \approx 156 - 24 \times 1.732 = 156 - 41.57 = 114.43$$ 12. **Calculate $GE$:** $$GE \approx \sqrt{114.43} \approx 10.7$$ 13. **Since $FG$ is the radius of the cone's base, and $GE$ is part of the base circle, $FG = GE$:** 14. **Check options:** Calculate each option numerically: - $4\sqrt{2} \approx 5.66$ - $6 = 6$ - $6\sqrt{2} \approx 8.49$ - $6\sqrt{3} \approx 10.39$ - $6\sqrt{5} \approx 13.42$ 15. **Closest to $GE \approx 10.7$ is $6\sqrt{3} \approx 10.39$ cm.** **Answer: $FG = 6\sqrt{3}$ cm (Option D).**