Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
a = ½√(2²+2²)
a = √(2) units.
b² = (√(2))²+(√(2))²
b = 2 units.
c² = 3²+1²
c = √(10) units.
d = (135+atan(⅓))°
e² = 10+2-2√(20)cos(135+atan(⅓))
e = 2√(5) units.
(a) Calculating tan alpha exactly. Let alpha be x.
(√(2)/sinx) = (2√(5)/sin(135+atan(⅓)))
x = asin(√(2)/10)
It implies;
tanx is;
tan(asin(√(2)/10))
= ⅐
(b) Calculating Green Area exactly.
(f/sin(45+atan(⅓))) = (√(10)/sin(180-asin(⅒√(2))-(45+asin(⅓)))
f = ⅓(4√(5)) units.
g = (45-asin(⅒√(2)))°
h² = 10+(80/9)-(8√(50)/3)cos(asin(√(2)/10))
h = ⅓(√(2)) units.
i² = (2/9)+(20)-(4√(10)/3)cos(atan(⅓))
i = ⅓√(146) units.
(√(146)/3sin(atan(⅓))) = (√(2)/3sinj)
j = asin(1/√(730))°
j = 2.12109639666°
k = (45+asin(1/√(730)))°
l = (90+asin(1/√(730)))°
(√(10)/sin (90+asin(1/√(730)))) = (m/sin (45+asin(1/√(730))))
m = 2.31888531 units.
Area Green exactly is;
Area triangle with height ⅓(4√(5)) units and base 2.31888531sin(45-asin(⅒√(2)))
= ½*2.31888531*⅓*4*√(5)sin(45-asin(⅒√(2)))
= 2.07407407407 square units.
= 56/27 square units.
Calculating length red to 2 decimal places.
n = 180-45-atan(⅓)
n = (135-atan(⅓))°
(o/sin(45)) = (2/sin(135-atan(⅓)))
o = ½√(10) units.
p² = (½√(10))²-√(2)²)
p = ½√(2) units.
tan22.5 = q/√(2)
q = (2-√(2)) units.
r = 0.69973662616 units.
s = 180-(45-2.12109639666)-45
s = 92.12109639666°
(√(10)/sin92.12109639666) = (t/sin(42.87890360334))
t = 2.153250645 units.
u² = 2.153250645²+(5/2)-2*2.153250645*0.5√(10)cos45
u = 1.52370388376 units.
(1.52370388376/sin45) = (2.153250645/sinv)
v = 87.79740183627°
w = 180-v
w = 92.20259816373°
x = ½(135-atan⅓)+w
x = 150.48512375226°
It implies;
Red length to 2 decimal places is;
(0.69973662616²+1.52370388376²-2*1.52370388376*0.69973662616cos150.48512375226)^(½)
= 2.16031565285 units.
≈ 2.16 units.
Therefore;
(a) is ⅐
(b) is 56/27 square units.
Red length is 2.16 units.
