2023 AMC 12B Problems/Problem 21

Solution

We augment the frustum to a circular cone. Denote by $O$ the apex of the cone. Denote by $A$ the bug and $B$ the honey.

By using the numbers given in this problem, the height of the cone is $6 \sqrt{3}$ . Thus, $OA = 12$ and $OB = 6$ .

We unfold the lateral face. So we get a circular sector. The radius is 12 and the length of the arc is $12\pi$ . Thus, the central angle of this circular sector is $180^\circ$ .

Because $A$ and $B$ are opposite in the original frustum, in the unfolded circular cone, $\angle AOB = \frac{180^\circ}{2} = 90^\circ$ .

Notice that a feasible path between $A$ and $B$ can only fall into the region with the range of radii between $OB = 6$ and $OA = 12$ . Therefore, we cannot directly connect $A$ and $B$ and must make a detour. Denote by $AC$ a tangent to the circular sector with radius 6 that meets it at point $C$ . Therefore, the shortest path between $A$ and $B$ consists of a segment $AC$ and an arc from $C$ to $B$ .

Because $OA = 12$ , $OC = 6$ and $\angle OCA = 90^\circ$ , we have $AC = \sqrt{OA^2 - OC^2} = 6 \sqrt{3}$ and $\angle AOC = 60^\circ$ . This implies $\angle COB = \angle AOB - \angle AOC = 30^\circ$ . Therefore, the length of the arc between $C$ and $B$ is $OB \cdot \pi \cdot \frac{\angle COB}{180^\circ} = \pi$ . Therefore, the shortest distance between $A$ and $B$ is \boxed{\textbf{(E) } 6 \sqrt{3} + \pi}.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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2023 AMC 12B Problems/Problem 21

Solution