Mathematics Question and Solution
Calculating y, the red required angle.
It implies;
½*πr² = 32π
r = √(64)
r = 8 cm.
r is the radius of the half circle.
2r = 16 cm, the diameter of the half circle.
Let x be the radius of the inscribed circle.
a = (8-x) cm.
Calculating x.
2x² = (8-x)²
2x² = 64-16x+x²
x²+16x-64 = 0
(x+8)² = 64+(8)²
(x+8)² = 128
x = -8±√(128)
x = -8±8√(2)
It implies;
x ≠ (-8-8√(2)) cm.
x = (8√(2)-8) cm.
x = 3.31370849898 cm.
Again, x is the radius of the inscribed circle.
b = 8+x
b = 8+3.31370849898
b = 11.31370849898 cm.
tanc = 3.31370849898/11.31370849898
c = atan(3.31370849898/11.31370849898)
c = 16.3249499369°
cos16.3249499369 = d/11.31370849898
d = 11.31370849898cos16.3249499369
d = 10.8575735127 cm.
e² = 16²+10.8575735127²-2*16*10.8575735127cos16.3249499369
e = 6.36021930932 cm.
(6.36021930932/sin16.3249499369) = (16/sinf)
f = asin(0.70710678118)
f = 135°
It implies, the required red angle is;
90+y = f
90+y = 135
y = 135-90
y = 45°
