Mathematics Question and Solution
Calculating area of the red inscribed circle.
Let a be the radius of the blue ascribed circle.
πa² =144π
a² = 144
a = 12 cm.
b = 2a
b = 24 cm.
b is the diameter of the blue ascribed circle.
2c² = 24²
c² = 12*24
c = 12√(2) cm.
c is the side length of the inscribed square.
Let y be the radius of the red inscribed circle.
d = 2y cm.
d is the diameter of the red inscribed circle.
2e² = (2y)²
e² = 2y²
e = √(2)y cm.
f = ½(e)
f = ½√(2)y cm.
g = 12√(2)-f
g = ½(24√(2)-√(2)y) cm.
h = ½(12√(2))-½√(2)y
h = ½(12√(2)-√(2)y) cm.
j = (6√(2)+y) cm.
Calculating y.
(6√(2)+y)² = (½(12√(2)-√(2)y))²+(½(24√(2)-√(2)y))²
72+12√(2)y+y² = ¼(288-48y+2y²)+¼(1152-96y+2y²)
288+48√(2)y+4y² = 288-48y+2y²+1152-96y+2y²
48√(2)y+4y² = 4y²-144y+1152
48√(2)y = 1152-144y
6√(2)y = 144-18y
2√(2)y = 72-9y
(2√(2)+9)y = 72
y = 72/(2√(2)+9)
y = 6.08703078107 units.
Again, x is the radius of the red inscribed circle.
Therefore, area red circle is;
πy²
= π(6.08703078107)²
= 116.402114222 cm²
Calculating cosx.
Recall.
j = (6√(2)+y)
And y = 6.08703078107 cm.
j = 14.5723121553 cm.
Recall Again.
h = ½(12√(2)-√(2)y)
And y = 6.08703078107 cm.
h = ½(12√(2)-√(2)*6.08703078107)
h = 4.18110063165 cm.
Therefore;
cosx = h/j
cosx = (4.18110063165/14.5723121553)
cosx = 0.2869208803
