A lampshade is the lateral surface of a frustum of a right circular cone with height $8\sqrt2$ inches, top diameter $4$ inches, and bottom diameter $12$ inches. A bug is at the bottom rim and there is a glob of honey on the top rim at the spot farthest from the bug. The bug crawls to the honey along the surface. What is the length of its shortest path, in inches?

1. The slant height of the frustum is $\sqrt{(6-2)^2+(8\sqrt2)^2}=12$. Extending to a full cone, similar triangles give apex-to-top slant $L'$ and apex-to-bottom slant $L$ with $L'/L=2/6$ and $L-L'=12$, so $L=18$ and $L'=6$.
2. Unroll the cone: the sector has radius $18$ and angle $\frac{2\pi\cdot6}{18}=120^\circ$. The bug and the honey are halfway around the cone from each other, i.e. $60^\circ$ apart in the unrolled sector; the bug is at distance $18$ from the apex and the honey at distance $6$.
3. By the Law of Cosines the straight segment has length $\sqrt{18^2+6^2-2\cdot18\cdot6\cos60^\circ}=\sqrt{252}$ (and this segment stays between the rims of the unrolled frustum).
4. The shortest path is $\boxed{6\sqrt7}$ inches.