Mathematics Question and Solution
Calculating yellow area.
Let x be the radius of the big inscribed circle.
Let y be the radius of the small inscribed circle.
a = (1+x) units.
b = (1-x) units.
Calculating x.
a² = 2b²
(1+x)² = 2(1-x)²
1+2x+x² = 2-4x+2x²
x²-6x+1 = 0
x²-6x+(-6/2)² = -1+(-6/2)²
x²-3x-3x+9 = -1+9
(x-3)² = 8
x = 3±2√(2)
It implies;
x = 3-2√(2)
x = 0.17157287525 units.
Again, x is the radius of the inscribed big circle.
c = x+y
c = (0.17157287525+y) units.
d = x-y
d = (0.17157287525-y) units.
e²+d² = c²
e² = (0.17157287525+y)²-(0.17157287525-y)²
e² = y²+0.3431457505y+0.02943725152-(y²-0.3431457505y+0.02943725152)
e² = 0.686291501y
e = √(0.686291501y) units.
f = 1-x-e
f = 1-0.17157287525-√(0.686291501y)
f = (0.82842712475-√(0.686291501y)) units.
g = (1+y) units.
h = (1-y) units.
Calculating y.
g² = f²+h²
(1+y)² = (0.82842712475-√(0.686291501y))²+(1-y)²
Therefore;
y = 0.08578643762 units.
Again, y is the radius of the small inscribed circle.
It implies, yellow area is;
Area square with side length 1 unit - Area quarter circle with radius 1 unit - Area circle with radius 0.17157287525 units - Area circle with radius 0.08578643762 units.
(1*1)-(¼*1*1*π)-(0.17157287525²π)-(0.08578643762²π)
= 0.09900202019 square units.
