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Question Number 16592 by chux last updated on 24/Jun/17
please what does the question mean  by the overlapping portion of A   and B.
pleasewhatdoesthequestionmeanbytheoverlappingportionofAandB.
Commented by chux last updated on 24/Jun/17
Answered by ajfour last updated on 24/Jun/17
Commented by ajfour last updated on 24/Jun/17
 Overapping area is the area  common to circles A and B.  Let  (h,0) be the center of circle B.   sin (x°/2)=(h/2)  ⇒  h=2sin (x°/2)  eqn of circle A:    (x+h)^2 +y^2 =1  eqn of circle B:    (x−h)^2 +y^2 =1  let circle A intersect x axis at (a,0).  then   a+h =1  or    a=1−h  ...(i)   overlapping area is two times the  area between circle A and y-axis.   upper portion of circle A has eqn.   y= (√(1−(x+h)^2 ))   let required area is S.   S=4∫_0 ^(  a) (√(1−(x+h)^2 )) dx    = 4{((x+h)/2)(√(1−(x+h)^2 )) +                                     (1/2)sin^(−1) (x+h) }∣_0 ^a      = 4(((a+h)/2))(√(1−(a+h)^2 ))−((4h)/2)(√(1−h^2 ))                 +(4/2)sin^(−1) (a+h)−(4/2)sin^(−1) h   but a+h=1  and  h=2sin (x°/2)   S = 0−2h(√(1−h^2 ))+π−2sin^(−1) h      = π−2h(√(1−h^2 ))−2sin^(−1) h .                   where h= 2sin (x°/2) .
OverappingareaistheareacommontocirclesAandB.Let(h,0)bethecenterofcircleB.sin(x°/2)=h2h=2sin(x°/2)eqnofcircleA:(x+h)2+y2=1eqnofcircleB:(xh)2+y2=1letcircleAintersectxaxisat(a,0).thena+h=1ora=1h(i)overlappingareaistwotimestheareabetweencircleAandyaxis.upperportionofcircleAhaseqn.y=1(x+h)2letrequiredareaisS.S=40a1(x+h)2dx=4{x+h21(x+h)2+12sin1(x+h)}0a=4(a+h2)1(a+h)24h21h2+42sin1(a+h)42sin1hbuta+h=1andh=2sin(x°/2)S=02h1h2+π2sin1h=π2h1h22sin1h.whereh=2sin(x°/2).
Commented by chux last updated on 24/Jun/17
thanks alot sir.
thanksalotsir.
Commented by chux last updated on 24/Jun/17
please is it possible to get the  value of x.
pleaseisitpossibletogetthevalueofx.
Commented by ajfour last updated on 24/Jun/17
that is to be chosen, in question  it said 0°≤x≤60°  , had it said    x=30° or any other value, area  of overlap shall be obtained then  from the derived formula, or if the  area of overlap be given the angle  x can be calculated; they are  interdependent.
thatistobechosen,inquestionitsaid0°x60°,haditsaidx=30°oranyothervalue,areaofoverlapshallbeobtainedthenfromthederivedformula,oriftheareaofoverlapbegiventheanglexcanbecalculated;theyareinterdependent.
Commented by chux last updated on 24/Jun/17
ok sir
oksir

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