Mathematics Question and Solution
Total area of the ascribed composite plane shape is;
(¼*12*12π) + (½*6*6π)
= 36π+18π
= 54π cm²
Calculating the inscribed shaded area.
Let re be the radius of the inscribed shaded half circle.
a = (12-x) cm.
Therefore;
(12-x)² = r²+x²
144-24x+x² = r²+x²
r² = (144-24x) cm.
It implies;
12² = (12-x)²+r²
Notice.
r² = (144-24x) cm.
144 = 144-24x+x²+(144-24x)
0 = -24x+x²+144-24x
x²-48x+144 = 0
(x-24)² = -144+(-24)²
(x-24)² = 576-144
x = 24±√(432)
Therefore;
x ≠ 24+√(432) cm.
x = 24-√(432) cm
x = 12(2-√(3)) cm
Recall.
r² = (144-24x) cm.
And x = 12(2-√(3)) cm.
r² = 144-24*12(2-√(3))
r² = 144-576+288√(3)
r² = 288√(3)-432
r = √(288√(3)-432)
r = 12√(2√(3)-3) cm.
r = 8.1750004636 cm.
Therefore;
Shaded Area ÷ Total Area is;
(0.5π*8.1750004636²)÷(54π)
= 0.61880215352
= ⅓(4(2√(3)-3) exactly in fraction.
