AMC 12 2006a Problem 12

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

size(7cm); pointpen = black; pathpen = linewidth(0.7);
D(CR((0,0),10));
D(CR((0,0),9.5));
D(CR((0,-18.5),9.5));
D(CR((0,-18.5),9));
MP("$\vdots$",(0,-31),(0,0));
D(CR((0,-39),3));
D(CR((0,-39),2.5));
D(CR((0,-43.5),2.5));
D(CR((0,-43.5),2));
D(CR((0,-47),2));
D(CR((0,-47),1.5));
D(CR((0,-49.5),1.5));
D(CR((0,-49.5),1.0));

D((12,-10)--(12,10)); MP('20',(12,0),E);
D((12,-51)--(12,-48)); MP('3',(12,-49.5),E);

$\mathrm{(A) \ } 171 \mathrm{(B) \ } 173 \mathrm{(C) \ } 182 \mathrm{(D) \ } 188 \mathrm{(E) \ } 210$