Mathematics Question and Solution
a = (1+x) units.
a is the diameter of the semi circle.
b = ½(a)
b = ½(1+x) units.
b is the radius of the semi circle.
c = x units.
c is the radius of the quarter circle.
Notice!
A-B = π
A = B+π
B = A-π
It implies;
Calculating x.
It is;
Area semi circle with radius ½(1+x) units - Area B + Area A - ½(area quarter circle with radius x units) = ½(area quarter circle with radius x units)
½(½(1+x))²π-B+A-½*¼(x²)π = ½*¼(x²)π
⅛(1+2x+x²)π-(A-π)+A-⅛(x²)π = ⅛(x²)π
⅛π+¼(x)π+π = ⅛(x²)π
⅛+¼(x)+1 = ⅛(x²)
⅛(x²)-¼(x)-(9/8) = 0
x²-2x-9 = 0
Resolving the above quadratic equation via completing the square approach to get x, radius of the quarter circle.
(x-1)² = 9+(-1)²
(x-1)² = 10
x-1 = ±√(10)
x = 1±√(10)
It implies;
x ≠ 1-√(10)
x = 1+√(10) units.
x = 4.1622776602 units.
