PUMaC 2008 · 团队赛 · 第 7 题

PUMaC 2008 — Team Round — Problem 7

(5 points) The graphs of the following equations divide the xy plane into some number of regions. 2 4 + ( x + 2) y = x 2 2 ( x + 2) + y =16 Find the area of the second smallest region. 2 3 3

解析

The graphs of the following equations divide the xy plane into some number of regions. 2 4 + ( x + 2) y = x 2 2 ( x + 2) + y =16 Find the area of the second smallest region. ( ANS: 4 π + 8 Rewriting, we find: ( x + 2) y = ( x − 2)( x + 2) to get x = − 2 or y = x − 2. The second equation is a circle centered at ( − 2 , 0) with radius 4. The second smallest region is given by ( x, y ) with 2 2 x > − 2, y > x − 2, and ( x + 2) + y < 16. This region can be divided into a quarter circle and an 1 isoceles right triangle. Thus the area is (1 / 4) π (4)2 + (4)2 = 4 π + 8. CB: JVP) 2