Area of stage will be: STAGE AREA= 3r x 2r = 6r^2 From the description and diagram
The LIT area will be
pi r^2 + (pi r^2 - 2 x (segment of overlap) ) (1)
Segment of overlap is portended by an angle of 120 degrees
120/360 (pi r^2) = 1/3 pi r^2 is the area of the SECTOR portended by the 120 degrees
we will have to SUBTRACT TWO TIMES the area of a triangle with hypotenuse = r and 1 leg = 1/2R to find the area of the SEGMENT
Using Pythagorean theorem to find the other leg r^2 - (1/2r)^2 = (other leg)^2
Other leg = sqrt(r^2-(1/2r^)^2 ) = sqrt(r^2-1/4r^2) = sqrt(3/4r^2)= r (sqrt3)/2
So after all of this, the area of this RIGHT triangle is 1/2 (Leg1 x Leg2) = 1/2 ( 1/2 r x r (sqrt3)/2) = 1/2 ( r^2 (sqrt3)/4) = r^2 sqrt3 / 8
NOW,(hang in there.....this get's long) TWO times this triangle is ( r^2sqrt3) /4
Area of SECTOR - TWO TIMES TRIANGLE = area of SEGMENT = 1/3 pi r^2 -( r^2 sqrt 3 ) /4 (2)
Getting there: from (1) the LIT area of the stage is pi r^2 + pi r^2 - 2 (1/3 pir^2- (r^2sqrt3)/4)
2 pi r^2 - 2/3 pi r^2 - r^2 (sqrt3)/2 = 4/3 pi r^2 -r^2 (sqrt3)/2
Now the PERCENTAGE of the stage is LIT area/ stage area 4/3 pi r^2 -r^2 (sqrt3)/2 / 6r^2
cancel out all of those r^2 to get: ( 4/3 pi - (sqrt3)/2 ) / 6 = 55.4% of the stage is lit 55.4 percent change flowers will land in the light
b) we found the are of the SECTOR to be (above--- (2) ) 1/3 pi r^2 -( r^2 sqrt 3 ) /4 the are of over lap is TWO times this
so the are of stage in overlapping light is 2 x sectors/ stage area = 2(1/3 pi r^2 -( r^2 sqrt 3 ) /4) / 6r^2 cancel out all of th r^2 to get
(2/3pi -( sqrt3)/2) / 6 = 20.47 percent chance of landing in overlap area
Wow.....there was a LOT of calcs to get this answer......hope I didn't mess up in there somewhere ! Anyone else get same answers???
