Mathematics Question and Solution
Calculating Green Area.
Let x be the equal inscribed angles.
Let a be the radius of the sector.
It implies;
b² = 2a²-2a²cosx --- (1).
b² = 6²+9²-2*6*9cosx
b² = 117-108cosx --- (2).
Equating (1) and (2).
2a²-2a²cosx = 117-108cosx
a²(2-2cosx) = 117-108cosx
a² = (117-108cosx)/(2-2cosx) --- (3).
c = a+a
c = 2a units.
It implies;
(2a)² = 6²+b²
4a² = 36+117-108cosx
4a² = 153-108cosx
a² = ¼(153-108cosx) --- (4).
Calculating x, equating (3) and (4).
(117-108cosx)/(2-2cosx) = ¼(153-108cosx)
Let cosx = d.
(117-108d)/(2-2d) = ¼(153-108d)
(117-108d)/(1-d) = ½(153-108d)
234-216d = 153-108d-153+108d²
81 = -45d+108d²
9 = 12d²-5d
12d²-5d-9 = 0
(d-(5/24))² = (9/12)+(-5/24)²
(d-(5/24))² = 0.79340
(d-0.20833)² = 0.79340
d = 0.20833±√(0.79340)
d = 0.20833-0.89073
Therefore;
d ≠ 0.20833+0.89073
d ≠ 1.09906
d = 0.20833-0.89073
d = -0.6824
And d = cosx
It implies;
x = acosd
x = acos(-0.6824)
x = 133.031473376°
Again, x is the the equal inscribed angles.
Calculating a, radius of the sector using (4).
At (4).
a² = a² = ¼(153-108cosx)
And x = 133.031473376°
a = √(0.25(153-108cos133.031473376))
a = 7.52826673279 units.
It implies;
Area Green is;
(133.031473376π*7.52826673279²)/360
= 65.7948300151 square units.
