Mathematics Question and Solution
Calculating Area Yellow.
Let a be the radius of the ascribed quarter circle.
(2a)² = b²+24²
b = √(4a²-576) units.
c = b-7
c = (√(4a²-576)-7) units.
It implies;
Calculating a.
(√(4a²-576)-7) ~ 2a
a ~ √(4a²-576)
Cross Multiply.
2a² = 4a²-576-7√(4a²-576)
7√(4a²-576) = 2a²-576
49(4a²-576) = (2a²-576)²
196a²-28224 = 4a⁴-2304a²+331776
4a⁴-2500a²+360000 = 0
a⁴-625a²+90000 = 0
(a²-½(625))² = -90000+(-0.5(625))²
(a²-½(625))² = ¼(30625)
a² = ½(625)±½√(30625)
a² = ½(625)±½(175)
a² = ½(625±175)
Therefore;
a² = ½(450) = 225 units.
Or
a² = ½(800) = 400 units.
It implies;
a ≠ √(225) = 15 units.
a = √(400) = 20 units.
cosd = 12/20
d = acos(3/5)°
e = (90-d)
e = (90-acos(3/5))°
Therefore, area S, yellow area is;
Area triangle with height 24 units and base 20sin(acos(3/5)) cm + Area triangle with height 20 units and base 7sin(90-acos(3/5))
= 192+42
= 234 square units.
