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Shifted regions (Posted on 2019-02-27) Difficulty: 2 of 5
A circle is centered at the origin with a radius of 5. Two lines are drawn that divide the circle into four regions, the first is the line y=-1, and the other is x=-3. Find the area of the largest region.

No Solution Yet Submitted by Danish Ahmed Khan    
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solution - math and computer | Comment 3 of 4 |
I finally fixed the trig. in my original posting and finished the puzzle as I had started it, and all the work is documented in that post. Bottom line:

A=41.645

I wrote two codes: One to produce the analytic solution I give, and a second using a simulation to check the result. They agree.

Perhaps where Charlie went wrong was approximating half sections using triangles.

Analytic Solution

rabbit-3:~ lord$ sec

  region area =    41.6451340    

rabbit-3:~ lord$ more sec.f

        program sec

        implicit none

        real pi,as1,as3,area,R,a2,a4,

        1 t1,t2,a,b,areatot,ang1,ang2

        pi = 3.14159

        R=5

        areatot=pi*25

        a=1

        b=3

        t1=2*acos(a/R)

        t2=2*acos(b/R)

        as1 = (1/4.)*R**2*(pi-t1+sin(t1))

        as3 = (1/4.)*R**2*(pi-t2+sin(t2))

        a2 = (1/4.)*areatot

        a4 = 3

        area = as1 + a2 + as3 + a4

        print*,' region area = ',area

        end


Simulation Solution


rabbit-3:~ lord$ sector

   41.6415939    

   41.6478386    

   41.6486015    

   41.6469307    

   41.6471672    

   41.6396904    

   41.6485176    

   41.6417007    

   41.6469078    

   41.6474457    

 ave    41.6456413   

rabbit-3:~ lord$ more sector.f

        program sector

        implicit none

        integer iseed,i,j,totc,tots

        real xx,yy,areac,areas,r2,ave

        ave=0

        do j=1,10

        iseed=time8()

        call srand(iseed)

        totc=0

        tots=0


                do i=1,10**8

1               xx=10*rand()-5

                yy=10*rand()-5

                r2=xx**2+yy**2

                if(r2.gt.25)go to 1

                totc=totc+1

                if(xx.ge.-3..and.yy.ge.-1.)tots=tots+1

                enddo

        areac=3.14159*25

        areas=(1.*tots/totc)*areac

        print*,areas

        ave=ave+areas

        enddo

        ave=ave/10

        print*,'ave ',ave

        end     


Edited on February 28, 2019, 11:40 am
  Posted by Steven Lord on 2019-02-28 11:20:57

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