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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 n number of circles can be painted, then the unpainted area available in annular space is ______.
Area of a circle with radius a = ?a2Area of a circle with radius b = ?b2The diameter of the shaded circle = (b - a) Area of shaded circle = \(\pi(\frac{b-a}{2})^2 = \pi \frac{(b-a)^2}{4}\)Painted area = total area of shaded circle = \(n \pi \frac{(b-a)^2}{4}\)Area of angular space = ?b2 “ ?a2 = ? (b2 “ a2 )Area of angular space = Painted area + Unpainted areaUnpainted area = Area of angular space “ Painted areaUnpainted area = \(\pi (b^2 - a^2) -n \pi \frac{(b-a)^2}{4}\)Unpainted area = \(\pi[(b^2 -a^2) - \frac{n}{4}(b-a)^2]\)
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