AMC 12 2008a Problem 22

A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?

unitsize(4mm);
defaultpen(linewidth(.8)+fontsize(8));
draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-1.55,2.1),E);
label("$1$",(-0.5,3.8),S);

$\mathrm{(A)}\ 2\sqrt{5}-\sqrt{3} \mathrm{(B)}\ 3 \mathrm{(C)}\ \frac{3\sqrt{7}-\sqrt{3}}{2} \mathrm{(D)}\ 2\sqrt{3} \mathrm{(E)}\ \frac{5+2\sqrt{3}}{2}$