Mathematics Question and Solution
Calculating red area.
a = (1+x) units.
a is the diameter of the circle.
b = ½(a)
b = ½(1+x) units.
b is the radius of the circle.
c = ½(1-x) units.
Calculating x.
(½(1+x))² = (½(1-x))²+(½(1-x))²
¼(1+2x+x²) = ¼(1-2x+x²)+¼(1-2x+x²)
1+2x+x² = 1-2x+x²+1-2x+x²
1+2x+x² = 2-4x+2x²
x²-6x+1 = 0
(x-3)² = -1+(-3)²
(x-3)² = 8
x = 3±√(8)
It implies;
x ≠ 3+√(8) units.
x = 3-√(8) units.
x = 0.1715728753 units.
Recall.
b = ½(1+x) units.
And x = 0.1715728753 units.
b = ½(1+0.1715728753)
b = 0.5857864376
b is the radius of the circle.
d = 1-x
d = 1-0.1715728753
d = 0.8284271247 units.
Therefore, area red is;
Area half circle with radius 0.5857864376 units - Area triangle with height and base 0.8284271247 units respectively + Area trapezoid with parallel sides 1 unit and 0.1715728753 units respectively, and height 0.8284271247 unit + Area rectangle with length 1 unit and width 0.1715728753 units - Area half circle with radius 0.5857864376 units.
= (0.5π*0.5857864376²)-(0.5*0.8284271247²)+(0.5*0.8284271247(1+0.1715728753))+(1*0.1715728753)-(0.5π*0.5857864376²)
= 0.5390120844-0.3431457505+0.4852813742+0.1715728753-0.5390120844
= 0.313708499 square units.
