Mathematics Question and Solution
Calculating Angle alpha and the circumference of the inscribed circle.
Let alpha be x.
Notice.
32π cm² is the area of the ascribed half circle.
Calculating R, radius of the ascribed half circle.
½*πR² = 32π
R² = 64
R = 8 cm.
Calculating r, radius of the inscribed circle.
a = (8-r) cm.
It implies;
2r² = a²
2r² = (8-r)²
2r² = 64-16r+r²
r²+16r-64 = 0
(r+8)² = 64+(-8)²
r+8 = ±√(128)
r = -8±8√(2)
It implies;
r ≠ -8-8√(2)
r = 8√(2)-8
r = 8(√(2)-1) cm.
r is the radius of the inscribed circle.
b = 8+r
b = 8+8√(2)-8
b = 8√(2) cm.
Therefore;
tanx = r/b
x = atan((8√(2)-8)/8√(2))
x = atan((√(2)-1)/√(2))
x = atan(½(2-√(2))°
x = 16.3249499369°
Again, x is alpha, the required angle.
Circumference of the inscribed circle, 2πr is;
2*π*8(√(2)-1)
= 16π(√(2)-1) cm.
