Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
(a) Calculating Length Blue ÷ AB.
Let AB be 1 unit.
a² = 1²+1²-2cos108
a = 1.61803398875 units.
b² = 1.61803398875²-0.5²
b = 1.53884176859 units.
c = 0.5+b
c = 2.03884176859 units.
cos54 = 0.5/d
d = 0.85065080835 units.
It implies;
Length Blue (CD) is;
(CD)² = 0.85065080835²+2.03884176859²-2*2.03884176859*0.85065080835cos108
CD = 2.43974693473 units.
Therefore;
Length Blue ÷ Length AB to 2 decimal places is;
2.43974693473 ÷ 1
= 2.43974693473
≈ 2.44
(b) Calculating Area Orange ÷ Area Purple.
e² = 2-2cos54
e = 0.90798099948 units.
Area purple is;
Area triangle with height 1 unit and base sin54 units + Area triangle with height 0.90798099948 units and base sin99 units.
½*sin54 + ½*0.90798099948*sin99
= 0.40450849719 + 0.44840112333
= 0.85290962052 square units.
f = 360-108-90
f = 162°
g = ½(360-2(162))
g = 18°
cos18 = h/1
h = 0.9510565163 unit.
sin18 = i/1
i = 0.30901699437 unit.
j = 1+2h
j = 2.9021130326 units.
Area Orange is;
Area trapezoid with parallel sides 2.9021130326 units and 1 unit, and height 0.30901699437 unit.
= ½*0.30901699437*3.9021130326
= 0.60290962052 square units.
Therefore;
Area Orange ÷ Area Purple to 2 decimal places is;
0.60290962052 ÷ 0.85290962052
= 0.70688570749
≈ 0.71
Therefore;
(a) is 2.44
(b) is 0.71
