Mathematics Question and Solution
Calculating Shaded Area.
a² = 1²-(4k)²
a = √(1-16k²) units.
1 ~ √(1-16k²)
4k ~ b
Cross Multiply.
b = 4k√(1-16k²) units.
Again.
1 ~ √(1-16k²)
√(1-16k²) ~ c
Cross Multiply.
c = (1-16k²) units.
d = 1-c
d = 1-(1-16k²)
d = 16k² units.
e = 1-b
e = (1-4k√(1-16k²)) units.
Therefore;
(5k)² = (16k²)²+(1-4k√(1-16k²))²
25k² = 256k⁴+1-8k√(1-16k²)+16k²(1-16k²)
25k² = 256k⁴+1-8k√(1-16k²)+16k²-256k⁴
9k² = 1-8k√(1-16k²)
8k√(1-16k²) = 1-9k²
64k²(1-16k²) = 1-18k²+81k⁴
64k²-1024k⁴ = 1-18k²+81k⁴
1105k⁴-82k²+1 = 0
Therefore;
k ≠ √(0.0153846) = 0.12403467257 units.
k = √(0.0588235) = 0.2425355644 units.
It implies;
5k is;
= 5*0.2425355644
= 1.21267782201 units.
4k is;
= 4*0.2425355644
= 0.97014225761 units.
cosf = 0.97014225761/1
f = 14.0363007635°
g = 90-f
g = 75.9636992365°
Therefore, the required shaded area green is;
1²-(0.5*1*0.97014225761sin14.0363007635)-(0.5*1*0.97014225761sin75.9636992365)
= 1-0.1176475-0.470588
= 0.4117645 square units.
