Mathematics Question and Solution
Calculating green inscribed circle circumference.
Let y be the radius of the green inscribed circle.
Notice.
ABCD is a square.
Let x be the side length of the square which is also the radius of the inscribed quarter circle.
a = ½(x) m.
Calculating x.
x² = (½x)²+6²
x²-¼(x²) = 36
¾x² = 36
3x² = 36*4
x² = 48
x = √(48)
x = 4√(3) m.
Again, x is the radius of the inscribed quarter circle and also the side length of the square.
b = y+x
b = (y+4√(3)) m.
c = y-x
c = (y-4√(3)) m.
Calculating y.
2c² = b²
2(y-4√(3))² = (y+4√(3))²
2(y²-8√(3)+48) = y²+8√(3)+48
2y²-16√(3)+96 = y²+8√(3)+48
y²-24√(3)y+48 = 0
(y-12√(3))² = -48+(-12√(3))²
(y-12√(3))² = -48+432
(y-12√(3))² = 384
y = 12√(3)±√(384)
y = 12√(3)±8√(6)
It implies;
y ≠ 12√(3)+8√(6)
y = (12√(3)-8√(6)) m.
y = 1.18869174856 m.
Again, y is the radius of the inscribed green circle.
Therefore, circumference of the inscribed green circle is;
2πy
= 2(12√(3)-8√(6))π m.
= 7.46877052932 m.
