Mathematics Question and Solution
Calculating Area Yellow.
a = 2+20
a = 22 units.
b²+x² = 20²
b = √(400-x²) units.
cosy = b/20
cosy = √(400-x²)/20 --- (1).
10² = 22²+√(400-x²)²-2*22√(400-x²) cosy
100 = 484+400-x²-44√(400-x²)cosy
44√(400-x²)cosy = 784-x²
cosy = (784-x²)/(44√(400-x²)) --- (2).
Equating (1) and (2).
√(400-x²)/20 = (784-x²)/(44√(400-x²))
20(784-x²) = 44(400-x²)
5(784-x²) = 11(400-x²)
3920-5x² = 4400-11x²
11x²-5x² = 4400-3920
6x² = 480
x² = 80
x = √(80)
x = 4√(5) units.
x is the shortest length (base) of the inscribed yellow area.
Recall.
b = √(400-x²) units.
And x = 4√(5) units.
b = √(400-(4√(5))²)
b = √(400-80)
b = √(320)
b = 8√(5) units.
b is the shorter length (height) of the inscribed yellow area.
Therefore, area yellow, inscribed right-angled triangle with base x (4√(5) units) and height b (8√(5) units) is;
½*x*b
= ½*4√(5)*8√(5)
= 16*5
= 80 square units.
