MathDB

A semicircle of diameter

1

sits at the top of a semicircle of diameter

2

, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt));
filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy] (A)\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad (B)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad (C)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad(D)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi (E)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi

Three semicircles of radius

1

are constructed on diameter

AB

of a semicircle of radius

2

. The centers of the small semicircles divide

\overline{AB}

into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3;
pair A=(-2,0), B=(2,0);
fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle);
dot((0,0)); dot((-1,0)); dot((1,0));
draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray);
label("

A

",A,W); label("

B

",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy] (A)\ \pi\minus{}\sqrt3 \qquad (B)\ \pi\minus{}\sqrt2 \qquad (C)\ \frac{\pi\plus{}\sqrt2}{2} \qquad (D)\ \frac{\pi\plus{}\sqrt3}{2} (E)\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}

19

Problems(2)