Mathematics Question and Solution
Calculating Alpha.
Let it be x.
Let a be the radius of the ascribed quarter circle.
b = (a-√(2)) units.
c = (a-2√(2)) units.
Calculating a.
a ~ (a-√(2))
(a-2√(2)) ~ √(2)
Cross Multiply.
√(2)a = a²-2√(2)a-√(2)a+4
√(2)a = a²-3√(2)a+4
a²-4√(2)a+4 = 0
(a-2√(2))² = -4+(-2√(2))²
(a-2√(2))² = -4+8
(a-2√(2))² = 4
a-2√(2) = ±2
a = 2√(2)±2
It implies;
a ≠ 2√(2)-2 units.
a = 2√(2)+2 units.
Again, a is the radius of the ascribed quarter circle.
d = a-2√(2)
d = 2√(2)+2-2√(2)
d = 2 units.
It implies;
Calculating theta, angle x.
sinx = 2/(2√(2)+2)
x = asin(2/(2√(2)+2))
x = 24.4698005207°
Calculating green area.
e = 90-x
e = 90-24.4698005207
e = 65.5301994793°
Therefore, area green is;
½*2*(2√(2)+2)sin65.5301994793
= (2√(2)+2)sin65.5301994793
= 4.39473645387 square units.
Or
f²+2² = (2√(2)+2)²
f² = 8+8√(2)+4-4
f = √(8+8√(2)) units.
Green Area is;
½*2*√(8+8√(2))
= √(8+8√(2)) square units.
= 4.39473645387 square units.
