Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the radius of the semi circle be 1 unit.
cos20 = a/2
a = 1.87938524157 units.
cos20 = 1/b
b = 1.06417777248 units.
c = a-b
c = 0.81520746909 units.
Where c is AB, the side of the square.
Area Square is;
(0.81520746909)²
= 0.66456321766 square units.
Calculating AF.
sin20 = d/2
d = 0.68404028665 units.
0.68404028665 = e
1.87938524157 = 0.81520746909 cross multiply.
1.87938524157e = 0.81520746909*0.68404028665
e = 0.2967112535 units.
f² = 0.2967112535²+0.81520746909²
f = 0.86752566856 units.
sin70 = g/0.68404028665
g = 0.64278760969 units.
cos70 = h/0.68404028665
h = 0.23395555688 units.
i² = (0.5*0.23395555688)²+(0.64278760969)²
i = 0.65334501744 units.
j = atan(0.5*0.23395555688/0.64278760969)
j = 10.31410481552°
k = 70+j
k = 80.31410481552°
l² = 0.68404028665²+(0.5*0.64278760969)²-0.64278760969*0.68404028665cos20
l = 0.39752884221 units.
(0.39752884221/sin20) = (0.5*0.64278760969/sinm)
m = 16.05238873239°
n = 180-16.05238873259-(20-10.31410481552)
n = 154.26171608293°
(0.68404028665/sin154.26171608293) = (o/sin16.05238873239)
o = 0.43556334496 units
Therefore;
AF is
= √(0.43556334496²+0.81520746909²-2*0.81520746909*0.43556334496cos80.31410481552)
= 0.85720383162 units.
(a) AF ÷ Radius to 2 decimal places is;
0.85720383162 ÷ 1
= 0.85720383162
≈ 0.86
(b) Area Blue ÷ Area Square to 2 decimal places.
sin20 = p/2
p = 0.68404028665 units.
q = 0.81520746909-0.68404028665
q = 0.13116718244 units.
tan20 = r/0.81520746909
r = 0.2967112535 units.
sin20 = s/0.68404028665
s = 0.23395555688 units.
tan20 = t/1
t = 0.36397023427 units.
cos20 = 0.36397023427/u
u = 0.38732903315 units.
v = 0.81520746909-u
v = 0.42787843594 units.
Area Blue is;
0.5(0.42787843594+0.13116718244)*0.81520746909 - 0.5*0.81520746909*0.2967112535 + 0.5*0.23395555688*0.68404028665sin70
= 0.22786908183-0.12094061501+0.07519186659
= 0.18212033341 square units.
Area Blue ÷ Area Square to 2 decimal places is;
0.18212033341 ÷ (0.81520746909)²
= 0.27404516014
≈ 0.27
