Mathematics Question and Solution
Calculating yellow area.
a = ½(270)-90
a = 135-90
a = 45°
Notice.
CD = BD = 23 units.
Let b be the radius of the quarter circle.
c = 2b units.
c is the diameter of the circle.
d²+24² = (2b)²
d = √(4b²-576) units.
e = d-23
e = (√(4b²-576)-23) units.
f² = 2b²
f = √(2)b units.
Calculating b.
(√(4b²-576)-23)²+23² = (√(2)b)²
4b²-576-46√(4b²-576)+529+529 = 2b²
2b²+482 = 46√(4b²-576)
b²+241 = 23√(4b²-576)
(b²+241)² = 529(4b²-576)
b⁴+482b²+58081 = 2116b²-304704
b⁴-1634b²+362785 = 0
It implies;
b = 37 units.
Again, b is the radius of the quarter circle.
sing = (0.5*24)/37
g = asin(12/37)°
h = 2g
h = 2asin(12/37)°
j = 90-h
j = (90-2asin(12/37))°
Therefore, yellow area is;
Area triangle with height 37 units and base 37sin(2asin(12/37)) units + Area triangle with height 37 units and base 37sin(90-2asin(12/37)) units - Area triangle with height and base 23 units respectively.
= 0.5*37*37sin(2asin(12/37))+0.5*37*37sin(90-2asin(12/37))-0.5*23*23
= 420+540.5-264.5
= 960.5-264.5
= 696 square units.
