Mathematics Question and Solution
Calculating Area S.
Let x be the radius of the quarter circle.
a²+24² = (2x)²
a² = 4x²-576
a = √(4x²-576) units.
b = a-7
b = (√(4x²-576)-7) units.
Calculating x.
(√(4x²-576)-7) ~ 2x
x ~ √(4x²-576)
Cross Multiply.
2x² = 4x²-576-7√(4x²-576)
7√(4x²-576) = 2x²-576
7²(4x²-576) = (2x²-576)²
49(4x²-576) = 4x⁴-2304x²+331776
196x²-28224 = 4x⁴-2304x²+331776
4x⁴-2500x²+360000 = 0
x⁴-625x²+90000 = 0
It implies;
x = 20 units.
Again, x is the radius of the quarter circle.
Recall.
a = √(4x²-576) units.
And x = 20 units.
a = √(4*20²-576)
a = 32 units.
Recall Again.
b = a-7
b = 32-7
b = 25 units.
c²+20² = 25²
c² = 625-400
c = √(225)
c = 15 units.
Therefore, area S, yellow inscribed quadrilateral area is;
Area triangle with height 32 units and base 24 units - Area triangle with height 20 units and base 15 units.
½(32*24)-½(20*15)
= (16*24)-(10*15)
= 384-150
= 234 square units.
