Question

一根圆柱形木料的直径为 $12$ 英寸。用两个完全穿过木料的平面切割，从木料中切出一个楔形体。第一刀垂直于圆柱轴线，第二刀所在的平面与第一刀所在的平面成 $45^\circ$ 角。这两个平面的交线与木料恰好只有一个公共点。该楔形体的体积为 $n\pi$ 立方英寸，其中 $n$ 是正整数。求 $n$ 。

A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi,$ where $n$ is a positive integer. Find $n.$

答案

答案：216

知识点：圆柱体积对称性

难度评级：2300

解答：

把第一刀看作水平切割。两个切割平面的交线与木料恰好只有一个公共点，因此它是半径为 $6$ 的圆形截面的切线。于是楔形体位于整个圆盘上方：在切点处高度为 $0$ ；又因为第二刀成 $45^\circ$ ，到直径另一端时高度线性增加到 $12$ 。

将圆盘上的每一点与它关于圆心的对称点配对：楔形体在这两点上方的高度之和恰好为 $12$ 。因此两个楔形体副本可以拼成一个半径为 $6$ 、高为 $12$ 的圆柱，楔形体的体积为 $\frac{1}{2}\,\pi \cdot 6^2 \cdot 12 = 216\pi.$ 所以 $n = 216$ 。

Take the first cut as horizontal. The line where the two cutting planes meet touches the log at exactly one point, so it is tangent to the circular cross-section of radius $6.$ The wedge therefore stands over the entire disk: its height is $0$ at the tangent point and, because the second cut is at $45^\circ,$ it rises linearly to $12$ at the diametrically opposite point.

Pair each point of the disk with its mirror image through the center: the wedge's heights over the two points add to exactly $12.$ So two copies of the wedge fit together into a cylinder of radius $6$ and height $12,$ and the wedge's volume is $\frac{1}{2}\,\pi \cdot 6^2 \cdot 12 = 216\pi.$ Thus $n = 216.$

2003 AIME II 第 5 题

答案：216

难度评级：2300

解答：

其他年份的第 5 题