Circle covers square Mathematical Olympiad

Circles cannot be cut to cover!

The number of circles with a radius of 10 that can be arranged in a square with a side length of 100:

(100/20 )*(100/20)=25 (pieces), [5 in each row, ***5 rows, the four sides are tangent to the square, and the circle is tangent to the adjacent circle]

Coverage = 25*3.14*10*10/(100*100)=78.5%,

The area that still needs to be covered=100*100*(95%-78.5%)=1650,

The 4 blank areas between the circle and the square with side length 20=20*20-3.14*10*10=86,

Each blank area=86/4=21.5,

The blank area between four tangent circles = 21.5*4=86,

***There are 4*4=16 such blank areas,

Cover with 16 circles, the area that can be covered=16*86=1376,

The remaining area to be covered=1650-1376=274,

tangent and adjacent to the edge The blank area surrounded by two circles and sides = 2*21.5=43,

***There are such blank areas 4*4=16,

And we need to use 274/ 43=6.37≈7 (pieces), [take an integer]

That is, cover it with 7 circles, the area that can be covered = 7*43=301>274,

So it is necessary To make the coverage rate above 95%, at least 25+16+7=48 circles should be used.