A two dimensional solid is made by alternating circles with radius a and b such that the sides of the circles touch. The packing fraction is defined as the ratio of the area under the circles to the area under the rectangle with sides of length x and y . The ratio r = b / a for which the packing fraction is minimized is closest to:

Step-by-step solution

Area of rectangle = xy = 2 a ( 2 a + 2 b ) = 4 a ( a + b ) Area covered by circles = πa 2 + πb 2 = π a 2 + b 2 Packing fraction (P.F.) = π a 2 + b 2 4 a 2 + ab = πa 2 1 + b 2 a 2 4 a 2 1 + b a Putting r = b a P . F . = π 1 + r 2 4 ( 1 + r ) For minimum P.F., d ( P . F . ) dr = 0 or π 4 2 r ( 1 + r ) - 1 + r 2 ( 1 + r ) 2 = 0 ⇒ r 2 + 2 r - 1 = 0 or r = - 2 + 4 + 4 2 = 2 - 1 = 0 . 414 Answer is option (A)