Mathematics Question and Solution
Calculating Area S, Yellow Area.
a = 20 units.
a is the radius of the quarter circle.
b = 2a
b = 40 units.
c²+20² = d²
d = √(c²+400) units.
e = 7+d
e = (7+√(c²+400)) units.
Calculating c.
cosx = (7+√(c²+400))/40 --- (1).
cosx = 20/√(c²+400) --- (2).
Equating (1) and (2) to get c.
(7+√(c²+400))/40 = 20/√(c²+400)
Cross Multiply.
800 = 7√(c²+400)+c²+400
(400-c²)² = 49(c²+400)
160000-800c²+c⁴ = 49c²+19600
c⁴-849c²+140400 = 0
(c²-(849/2))² = -140400+(-849/2)²
(c²-(849/2))² = 39800.25
c² = (849/2)±√(39800.25)
c² = 424.5±199.5
c ≠ √(424.5+199.5)
c = √(424.5-199.5)
c = √(225)
c = 15 units.
Recall.
d = √(c²+400)
And c = 15 units.
d = √(15²+400)
d = 25 units.
f²+7² = 25²
f = √(25²-7²)
f = √(625-49)
f = √(576)
f = 24 units.
It implies, Area S, yellow area is;
0.5*7*24+0.5*20*15
= 84+150
= 234 square units.
