What is the maximum fractional coverage of the area of a circle by three, non-overlapping rectangles, all of which lie internal to the circle? (The rectangles can be of different sizes)
As a check, I ran my approach out to the bitter end. First I differentiated my area function and set it to zero. However the result was so horrific to solve, I instead opted to just test the function to see if it peaked at sqrt(2-2/sqrt(5)) as found by B. Smith. It did (see below). So his approach (extending the rotational symmetry of a circle to simplify the problem) worked beautifully.
program yyyo
implicit real*8 (a-z)
integer ib
x2b=sqrt(2.d0-2.d0/sqrt(5.d0))
xa=2.d0*(sqrt(5.d0)-1.d0)
do ib=52571,52576
b=ib/100000.
term1 = b*(1-b**2)**0.5d0
term2a = ( (b+ sqrt(b**2+8.d0) )/4.d0 )**2.d0
term2 = sqrt(1.d0 - term2a)
term3 = (b+ sqrt(b**2.d0+8.d0) )/4.d0 -b
a = 4.d0*(term1+term2*term3)
print*,x2b,2*b,xa,a
enddo
sqrt(2-2/sqrt(5)) trial height 2 (sqrt(5)-1) resulting area
1.05146222423 1.05141997337 2.47213595499 2.47213595356 1.05146222423 1.05144000053 2.47213595499 2.47213595460 1.05146222423 1.05146002769 2.47213595499 2.47213595499 1.05146222423 1.05148005485 2.47213595499 2.47213595474 1.05146222423 1.05149996280 2.47213595499 2.47213595385
Comparisons of trial heights and resulting areas. The peak occurs on the 3d line.
Edited on April 29, 2019, 4:19 pm
Confirming
