Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Calculating the exact hatched area.
Let x be the radius of the inscribed circle.
2x² = a²
a = √(2)x cm.
b = x+a
b = (x+√(2)x) cm.
c = 16+b
c = (16+x+√(2)x) cm.
c is the diagonal of the ascribed square.
Again.
c² = 2(16)²
c = 16√(2) cm.
Therefore, calculating x.
Equating the both c.
(16+x+√(2)x) = 16√(2)
(1+√(2))x = (16√(2)-16)
x = (48-32√(2)) cm.
x = 16(3-2√(2)) cm.
x = 2.74516600406 cm.
Again, x is the radius of the inscribed circle.
d = 16+x
d = 16+(48-32√(2))
d = (64-32√(2)) cm.
e = 16-x
e = 16-(48-32√(2))
e = (32√(2)-32) cm.
f = 45+90
f = 135°
Therefore, hatched area exactly is;
½(area triangle with height and base (32√(2)-32) cm) + Area rectangle with length (32√(2)-32) cm and width (48-32√(2)) cm - Area sector with radius 16 cm and angle 45° - Area sector with radius (48-32√(2)) cm and angle 135°
= ½(32√(2)-32)²+((32√(2)-32)*(48-32√(2)))-(45π*16²/360)-(135π(48-32√(2))²/360)
= ½(2048-2048√(2)+1024)+(1536√(2)-2048-1536+1024√(2))-32π-⅜(2304-3072√(2)+2048)
= 1024-1024√(2)+512-3584+2560√(2)-32π-864π+1152√(2)π-768π
= -2048+1536√(2)-1664π+1152√(2)π
= (1536√(2)-2048-1664π+1152√(2)π) cm²
= 14.8230009901 cm²
