Question

An infinite number of identical circular disc each of the radius half a tightly packed with such that the centres of the disc are at the integer values of coordinates x and y the ratio of the area of the uncovered patches to the total area is

Answer 1

(4 - π)/4 = 0.2146 is the ratio of the area of the uncovered patches to the total area

Step-by-step explanation:

Let say number of Disks are n²

Radius of Circle = 0.5

Diameter of circle = 2 * 0.5 = 1

Then size of Square box  = n * n

Area = n²

Number of Disks = n²

Radius = 0.5 = 1/2

Area of Disks  =  π *(1/2)² * n²   = n²π/4

area of the uncovered patches = n² - n²π/4

= n² (4 - π)/4

ratio of the area of the uncovered patches to the total area = (n² (4 - π)/4)/ n²

=  (4 - π)/4

= 0.2146

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