2010 AMC 8 #18
Source:
May 22, 2011
geometryrectangleratio
Problem Statement
A decorative window is made up of a rectangle with semicircles at either end. The ratio of to is . And is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle. [asy]
import graph; size(5cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(8); defaultpen(dps); pen ds=black; real xmin=-4.27,xmax=14.73,ymin=-3.22,ymax=6.8; draw((0,4)--(0,0)); draw((0,0)--(2.5,0)); draw((2.5,0)--(2.5,4)); draw((2.5,4)--(0,4)); draw(shift((1.25,4))*xscale(1.25)*yscale(1.25)*arc((0,0),1,0,180)); draw(shift((1.25,0))*xscale(1.25)*yscale(1.25)*arc((0,0),1,-180,0));
dot((0,0),ds); label("",(-0.26,-0.23),NE*lsf); dot((2.5,0),ds); label("",(2.61,-0.26),NE*lsf); dot((0,4),ds); label("",(-0.26,4.02),NE*lsf); dot((2.5,4),ds); label("",(2.64,3.98),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]