Mathematics Question and Solution
Let the small inscribed circle's radius be x.
x = 6 cm.
Let the big inscribed circle's radius be y.
y = 8 cm.
Let the ascribed semi circle's radius be z.
= 18 cm.
a² = 10²-8²
a = 6 cm.
b² = 12²-6²
b = 6√(3) cm
c = atan(z/b)°
c = atan(18/6√(3))°
c = atan√(3)°
c = 60°
d = atan(z/a)°
d = atan(18/6)
d = atan(3)°
(a) Angle theta exactly is;
Let it be e.
e = 180-c-d
e = 180-60-atan3
e = (120-atan3)°
e = (atan(⅓)+30)°
(b) Calculating Area Shaded to 1 decimal place cm².
f = atan(1/√(3)) = 30°
g = atan(4/3)°
h = (150-atan(4/3))°
i² = 2(18)²-2(18)²cos(150-atan(4/3))
i = 26.9353054 cm.
j = ½(180-(150-atan(4/3)))
j = 41.56505117708°
Area Shaded in cm² to 1 decimal place is;
½*18²sin(150-atan(4/3)) - ½*6²sin(96.86989764584) - (83.13010235416π*6²/360) - ½*8²sin(96.86989764584) - (83.13010235416π*8²/360) + ½(6*6√(3)) - (60π*6²/360) + ½(6*8) - (atan(4/3)π*8²/360)
= 160.83689233046 - 17.8707658145 - 26.1160918848 - 31.77025033688 - 46.4286077952 + 18√(3) - 18.84955592154 + 24 - 20.59203548139
= 54.38649963239 cm²
≈ 54.4 cm²
