The angle angleCAB sweeps out some fraction of o.A's full 360^@, and subsequently the same fractional part of o.A's circumference. To find what the fraction is, we can divide angleCAB by 360:
(mangleCAB)/360=55/360=11/72
So, angleCAB sweeps out 11/72 of the circle's circumference.
The circumference of any circle is equal to the product of pi and the diameter d of the circle, which is itself equal to twice the radius, 2r. Here, we're given r=AC=12, so the circle's circumference is pi*2(12)=24pi.
So, the length of the arc BC=11/72*24pi=(24pi*11)/72
24 and 72 both have the factor 24 in common, so we can reduce down to
(11pi)/3
And we're done.