PUMaC 2011 · 几何（A 组） · 第 1 题

PUMaC 2011 — Geometry (Division A) — Problem 1

[ 3 ] Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as nπ , where n is an integer. Find n .

解析

Note that the solid formed is a generalized cylinder. It is clear from the diagram that the area of the base of this cylinder (i.e., a vertical cross-section of the log) is composed of two semicircles of radius 3 and a part of an annulus. In the right triangle in the diagram, the hypotenuse is 4 and the vertical leg is 2. Thus, it is a 30-60-90 triangle, so the central angle in ◦ the annulus is 120 . Since the annular region has inner radius 1 and outer radius 7, the total 1 1 2 2 2 area is 2( π 3 ) + π (7 − 1 ) = 25 π . Hence the volume of the cylinder is 10 · 25 π = 250 π , so 2 3 the answer is 250 . Figure 1: Problem 1 diagram Figure 2: Problem 2 diagram