Mathematics Question and Solution
a² = 4²+3²
a = √(25)
a = 5 units.
a is the radius of the semi circle.
tanb = (3/4)
b = atan(3/4)°
c = 90-b
c = (atan(4/3))°
tan(atan(3/4)) = d/5
(3/4) = d/5
d = 3.75 units.
d = ¼(15) units.
tan(atan(4/3)) = e/5
(4/3) = e/5
e = ⅓(20) units.
Area 1 is;
Area triangle with height 5 units and base ¼(15) units - Area sector with radius 5 units and angle atan(3/4)°
= ½*5*¼(15)-(atan(3/4)π*25/360)
= ⅛(75)-(25atan(3/4)π/360)
= 25(135-atan(3/4)π)/360 square units.
Area 2 is;
Area triangle with height 5 units and base ⅓(20) units - Area sector with radius 5 units and angle atan(4/3)°
= ½*5*⅓(20)-(atan(4/3)π*25/360)
= ⅙(100)-(25atan(4/3)π/360)
= 25(240-atan(4/3)π)/360 square units.
Therefore;
Area 1 ÷ Area 2, exactly as a single fraction in its simplest form is;
(25(135-atan(3/4)π)/360)÷(25(240-atan(4/3)π)/360)
= (25(135-atan(3/4)π))/(25(240-atan(4/3)π))
= (135-atan(3/4)π)/(240-atan(4/3)π)
= 0.56 to 2 decimal places.
