Mathematics Question and Solution
Calculating green area.
a = (6+x) units.
a is the radius of the circle.
b = a-2
b = (6+x)-2
b = (4+x) units.
(6+x)² = (4+x)²+c²
36+12x+x² = 16+8x+x²+c²
c² = 20+4x
c = √(20+4x) units.
c' = (6+x)+(4+x)
c' = (10+2x) units.
d²+x² = (6+x)²
d² = 36+12x
d = √(36+12x) units.
Calculating x.
(10+2x) ~ √(36+12x)
√(20+4x) ~ x
Cross Multiply.
(10x+2x²) = √((36+12x)(20+4x))
100x²+40x³+4x⁴ = 720+144x+240x+48x²
100x²+40x³+4x⁴ = 720+384x+48x²
4x⁴+40x³+52x²-384x-720 = 0
x⁴+10x³+13x²-96x-180 = 0
It implies;
x = 3 units.
Recall.
a = (6+x) units.
And x = 3 units.
a = 6+3
a = 9 units.
Again, a is the radius of the circle.
e²+3² = 9²
e² = 81-9
e = √(72)
e = 6√(2) units.
f² = 6²+(6√(2))²
f² = 36+72
f = √(108)
f = 6√(3) units.
sing = (0.5*6√(3))/9
g = 35.2643896828°
h = 4g
h = 141.057558731°
Area Green is;
½(area sector with radius 9 units and angle 141.057558731°- area triangle with height 9 units and base 9sin141.057558731 units).
= ½(141.057558731π*9²/360 - ½*9*9sin141.057558731)
= ½(99.7077128046-25.4558441227)
= ½(74.2518686819)
= 37.125934341 square units.
