PUMaC 2012 · 几何（B 组） · 第 2 题

PUMaC 2012 — Geometry (Division B) — Problem 2

[ 3 ] A 6-inch-wide rectangle is rotated 90 degrees about one of its corners, sweeping out an area of 45 π square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.

解析

[ 3 ] A 6-inch-wide rectangle is rotated 90 degrees about one of its corners, sweeping out an area of 45 π square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. Solution: After setting up the problem, we can see that the swept-out area is equal to a quartercircle with the rectangles diagonal as its radius. Since 45 π square inches is the area of √ the quartercircle, the circle’s radius is 6 5 inches. Using the Pythagorean Theorem, we have √ that the length equals 180 − 36 = 12 inches. Problem contributed by Luke Paulsen