a-maths!!!!!1

(a)

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Aug07/Crazytrigo13.jpg

(b)

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Aug07/Crazytrigo14.jpg

a)S=OQ+OR
=lcos(π/4) +lcosΘ
=√2l/2 +lcosΘ

dS/dΘ=-lsinΘ
when dS/dΘ=0,
sinΘ=0
Θ=0rad. or 2π(rejected)
d^2S/dΘ^2=-lcosΘ
whenΘ=0, d^2S/dΘ^2<0
∴S=l(√2/2 +1) is the maximum value

b)A=0.5xOQxORxsin(π/4 +Θ)
=0.5x(√2l/2)x( lcosΘ)xsin(π/4 +Θ)
=√2l^2/4 xcosΘxsin(π/4 +Θ)

dA/dΘ=√2l^2/4x[(cosΘ)xcos(π/4 +Θ) + sin(π/4 +Θ)(-sinΘ)]
=√2l^2/4x[(cosΘ)xcos(π/4 +Θ) + sin(π/4 +Θ)(-sinΘ)]
=√2l^2/4xcos(2Θ+π/4)

When dA/dΘ=0,
√2l^2/4xcos(2Θ+π/4)=0
cos(2Θ+π/4)=0
2Θ+π/4= π/2 or 3π/2
2Θ=π/4 or 5π/4
Θ=π/8 or 5π/8(rejected)

WhenΘ=π/8,
d^2A/dΘ^2=√2l^2/4x-2sin(2Θ+π/4)
=-√2l^2/2xsin(2Θ+π/4)
<0
∴the maximum area=√2l^2/4 xcos(π/8)xsin(π/4 +π/8)
=0.653l^2 square unit.