Mathematics Question and Solution
Calculating Green Area.
Notice.
The composite plane shape is not drawn to scale.
Let x be the width of the ascribed rectangle.
Therefore;
x = the diameter of the inscribed small half circle, and also the radius of the inscribed quarter circle.
b = (1+x) units.
b is the length of the ascribed rectangle, and also the diameter of the big inscribed half circle.
c = ½(b)
c = ½(1+x) units.
c is the radius of the big inscribed half circle.
d = ½(x) units.
d is the radius of the small inscribed half circle.
e²+(½(x))² = (½(x+1))²
e² = ¼(x²)+¼(2x)+¼-¼(x²)
e² = ¼(2x+1)
e = ½√(2x+1) units.
Calculating x.
x ~ ½(1+x)
½(x) ~ ½√(2x+1)
Cross Multiply.
½(1+x) = √(2x+1)
¼(x²+2x+1) = 2x+1
x²+2x+1 = 8x+4
x²-6x-3 = 0
(x-3)² = 3+(-3)²
x = 3±√(12)
It implies;
x ≠ 3-2√(3) units.
x = 3+2√(3) units.
x = 6.46410161514 units.
Again, x is the diameter of the inscribed small half circle, and also the radius of the inscribed quarter circle.
Recall.
b = (1+x)
And x = 3+2√(3) units.
b = 1+3+2√(3)
b = (4+2√(3)) units.
b = 7.46410161514 units.
Again, b is the length of the ascribed rectangle, and also the diameter of the inscribed big half circle.
d = ½(x)
And x = 6.46410161514 units.
d = ½(6.46410161514)
d = 3.23205080757 units.
c = ½(b)
And b = 7.46410161514 units.
c = ½(7.46410161514)
c = 3.73205080757 units.
cosf = 3.23205080757/3.73205080757
f = 30°
g = f+90
g = 120°
h² = 6.46410161514²+3.23205080757²-2*6.46410161514*3.23205080757cos120
h = 8.55120266156 units.
(8.55120266156/sin120) = (3.23205080757/sinj)
j = 19.1066053506°
k = 2j
k = 38.2132107013°
l = 2(180-120-j)
l = 81.7867892988°
Area green is;
Area sector with radius 6.46410161514 units and angle 30°+Area sector with radius 3.73205080757 and angle 120°-Area triangle with height 3.73205080757 units and base 3.73205080757sin120 units-Area sector with radius 3.23205080757 units and angle 81.7867892988°+Area triangle with height 3.23205080757 units and base 3.23205080757sin81.7867892988 units-Area sector with radius 6.46410161514 units and angle 38.2132107013°+Area triangle with height 6.46410161514 units and base 6.46410161514sin38.2132107013 units.
= 10.9391852365+14.5855803153-6.03108891325-7.45567365031+5.16950478279-13.9340463448+12.9237619568
= 16.197223383 square units.
