jmc

Question

Three concentric circles centered at O have radii of 1, 2, and 3. Points B and C lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angles BOC, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of BOC in degrees? FIGURE

Accepted Answer

Let x=BOC. We see that the shaded region is the inner ring plus a sector x^ of the outer ring. The area of this in terms of x is ( 4 - )+x360 ( 9 - 4 ). This simplifies to 3 + x360(5 ). Also, the unshaded portion is comprised of the smallest circle plus the sector (360-x)^ of the outer ring. The area of this is + 360-x360(5 ). We are told these are equal, therefore + x360(5 ) = 3 + 360-x360(5 ). Solving for x reveals x=108.