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HMMT 二月 2001 · 几何 · 第 4 题

HMMT February 2001 — Geometry — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A circle has two parallel chords of length x that are x units apart. If the part of the circle included between the chords has area 2 + π , find x .
解析
  1. A circle has two parallel chords of length x that are x units apart. If the part of the circle included between the chords has area 2 + π , find x . Solution: Let C be the area of the circle, S be the area of the square two of whose edges are the chords, and A be the area of the part of the circle included between the chords. √ 2 π 2 2 The radius of the circle is x , so C = x , and S = x . Then the area A is the area 2 2 of the square plus one half of the difference between the areas of the circle and square: √ π 1+ C − S C + S 2 A 2 2 A = + S = = x , so x = = 2 . π 2 2 2 1+ 2