Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Calculating Blue Area Exactly.
Calculating Area Brown.
a = (180(12-2))/12
a = 1800/12
a = 300/2
a = 150°
a is the single interior angle of the regular polygon with 12 sides.
b = ½(360-300)
b = 30°
c = ½(360-240)
c = 60°
sin30 = d/4
d = 2 cm.
sin60 = e/4
e = 2√(3) cm.
f = 2e+4
f = (4+4√(3)) cm.
sin30 = g/4
g = 2 cm.
sin60 = h/4
h = 2√(3) cm.
j = 2g+f
j = 2(2)+(4+4√(3))
j = (8+4√(3)) cm.
Brown Area is;
½*2(4+4+4√(3))+½*2√(3)((4+4√(3))+ (8+4√(3)))+½*2 (8+4√(3))
= (8+4√(3))+√(3)(12+8√(3))+(8+4√(3))
= 2(8+4√(3))+24+12√(3)
= 16+8√(3)+24+12√(3)
= (40+20√(3)) cm²
Calculating Area Regular Hexagon Side Length.
Let it be x.
k² = 2x²-2x²cos120
k = √(3)x cm.
It implies;
2*½(x²sin120)+(√(3)x*x) = Area Brown
½√(3)x²+√(3)x² = (40+20√(3))
3√(3)x² = 80+40√(3)
x² = ⅑(80√(3))+⅓(40)
x² = ⅑(80√(3)+120)
x = √(⅑(80√(3)+120))
x = ⅓(2√(20√(3)+30) cm.
x = 5.35997579395 cm.
Again, x is the side length of the regular hexagon.
l = ½(x)
l = ½*⅓(2√(20√(3)+30)
l = ⅓√(20√(3)+30) cm.
l = 2.67998789697 cm.
Area Blue Exactly is;
½*2(⅓√(20√(3)+30))²sin120
= ½√(3)(⅓√(20√(3)+30))²
= ½√(3)*⅑(20√(3)+30)
= ⅑√(3)(10√(3)+15)
= ⅑(30+15√(3))
= ⅓(5√(3)+10) cm²
= 6.22008467928 cm²
= (a√(3)+2a)/b
Where;
a = 5
b = 3
