The figure below shows an annular ring with outer and inner radii as b and a, respectively. The annular space has been p...

Question

The figure below shows an annular ring with outer and inner radii as b and a, respectively. The annular space has been painted in the form of blue colour circles touching the outer and inner periphery of annular space. If maximum number of circles can be painted, then the unpainted area available in annular space is ______.

A. pi[(b^2 - a^2) - (b - a)^2/2]
B. pi[b^2 - a^2 - (b - a)^2]
C. pi[b^2 - a^2 + (b - a)^2/2]
D. pi[b^2 - a^2 + (b - a)^2]

Accepted Answer

Unpainted area = pi[(b^2 - a^2) - (b - a)^2/2] An annular ring has outer radius b and inner radius a. The annular width is (b - a). Blue circles are inscribed in the annular space, each touching both the outer and inner circles. Such a circle has radius r = (b - a)/2 and its centre is at distance (a + b)/2 from the common centre. The maximum number of such circles that fit depends on spacing, but the key result is that each such circle has area pi*((b-a)/2)^2. The unpainted area is the annular area minus the total area of all blue circles.