Mathematics Question and Solution
Let the bigger inscribed semi circle radius be 2 units.
Let the big inscribed semi circle radius be r.
Let the inscribed circle radius be q.
Calculating r.
(2+r)² = 2²+(4-r)²
4+4r+r² = 4+16-8r+r²
4r = 16-8r
12r=16
3r=4
r=4/3 unit.
Calculating q.
(2+(4/3))²=(2+q)²+((4/3)+q)²
(10/3)²=4+4q+q²+(16/9)+(8q/3)+q²
(100/9)-(16/9)-4=2q²+4q+(8q/3)
(84/9)-4=2q²+4q+(8q/3)
84-36=18q²+36q+24q
48=18q²+60q
6q²+20q-16=0
3q²+10q-8=0
Therefore;
q≠-4
q=⅔
Blue shaded area is;
Area quarter circle of radius 4 units - area semi circle of radius 2 units - area semi circle of radius (4/3) units - area circle of radius (⅔) units.
= ¼(16π)-½(4π)-½((16/9)π)-(4/9)π
= 4π-2π-(8/9)π-(4/9)π
= 2π-((4/3)π)
= ⅓(6π-4π)
= ⅔(π) square units.
The shaded fraction is;
Shaded area ÷ Area of the ascribed quarter circle.
= ⅔(π)÷¼(16π)
= ⅔(π)÷4π
= ⅙
