Question

一个灯罩的形状是直圆锥台的侧面。圆锥台的高为 $3\sqrt{3}$ 英寸，上底直径为 $6$ 英寸，下底直径为 $12$ 英寸。一只虫子在灯罩底边上，灯罩上边缘离虫子最远的位置有一团蜂蜜。虫子想爬到蜂蜜处，但必须留在灯罩表面上。它到蜂蜜的最短路径长度是多少英寸？

A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt{3}$ inches, its top diameter is $6$ inches, and its bottom diameter is $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?

$6+3\pi$

$6+6\pi$

$6\sqrt{3}$

$6\sqrt{5}$

$6\sqrt{3}+\pi$

答案：E

知识点：圆锥展开图（立体几何）切线

难度评级：2020

解答：

将圆锥台延伸成完整圆锥。由于半径为 $3$ 和 $6$ ，斜高带宽为 $6$ ，顶点到上边缘的斜距为 $6$ ，到下边缘的斜距为 $12$ 。下底周长 $12\pi$ 展开成半径 $12$ 、角度 $\dfrac{12\pi}{12}=\pi$ 的扇形。在这个展开图中把虫子放在 $(12,0)$ ；蜂蜜位于沿底边走半圈的位置，在展开图中半径为 $6$ 、角度为 $\tfrac{\pi}{2}$ 。连接它们的直弦会进入半径 $6$ 以内（不在表面上），所以测地线与半径 $6$ 的圆相切：切线长为 $\sqrt{12^2-6^2}=6\sqrt3$ ，切点角度为 $\tfrac{\pi}{3}$ ，之后沿半径 $6$ 的圆弧走过角度 $\tfrac{\pi}{6}$ ，弧长为 $6\cdot\tfrac{\pi}{6}=\pi$ 。最短路径为 $6\sqrt3+\pi$ 。

因此，正确答案是 E。

Extend the frustum to a full cone. Since the radii are $3$ and $6$ with slant band $6,$ the apex is slant distance $6$ from the top rim and $12$ from the bottom rim. The bottom circumference $12\pi$ unrolls to a sector of radius $12$ and angle $\dfrac{12\pi}{12}=\pi.$ Place the bug at $(12,0)$ in this pattern; the honey, halfway around the base, is at radius $6$ and angle $\tfrac{\pi}{2}.$ The straight chord between them passes within radius $6$ (off the surface), so the geodesic goes tangent to the circle of radius $6:$ the tangent has length $\sqrt{12^2-6^2}=6\sqrt3$ and touches at angle $\tfrac{\pi}{3},$ after which the path follows the arc of angle $\tfrac{\pi}{6}$ on radius $6,$ of length $6\cdot\tfrac{\pi}{6}=\pi.$ The shortest path is $6\sqrt3+\pi.$

Thus, the correct answer is E.

2023 AMC 12B 第 21 题

答案：E

难度评级：2020

解答：

其他年份的第 21 题