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Question-203392

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Question Number 203392 by professorleiciano last updated on 18/Jan/24
Commented by a.lgnaoui last updated on 19/Jan/24
Commented by a.lgnaoui last updated on 20/Jan/24
△OAB a^2 =41−40cos x a=(√(41−40cos x)) { ((4sin x= asin y)),((4cos x+acos y=5)) :} Aire(OAB)=(1/2)(4×5)sin x=10sin x Aire(OAD) = (1/2)OC×3sin x = (3/2)×((12)/5)sin x=((18)/5)sin x Aire[(OAB)+(OCD)]=((68)/5)sin x (1) Calcul de x h=(4+((12)/5))sin x=((32)/5)sin x Aire totale=(((3+5)h)/2)=4h =Aird(BCD)+ACD= Aire =2Aird (OCD)+A2+A3 =Aird Titale −OAB+OCD ((3ACsin x)/2)+((3BDsin y)/2)= ((128sin x)/5)−Aire(OAB)+Aire(OCD) ⇒Aire(OAB)−Aire(OCD) (2) =((128sin x)/5)−(3/2)(ACsin x+BDsin y) ((OD)/(BD))=((12/5)/(4+12/5))=((12)/(32))=(3/8) BD=(8/3)OD ((OD)/(OB))=(3/5) OD=(3/5)a { ((AC=4+((12)/5)=((32)/5))),((BD=(8/3)×(3/5)a=(8/5)a)) :} ((OAB)/(AOD))=((25)/9) ((OAB+AOD)/(OAB))=((34)/(25))(3) △ABC ABsin 𝛌=ACsin x bsin 𝛌=((32)/5)sin x b=(√(25+(((32)/5))^2 −64cos x)) ((32)/5)sin x=(√(((1649)/(25))−64cos x)) 32^2 cos^2 x−1600cos x+625=0 cos x=((25)/(32)) sin x=((√(399))/(32))≅((20)/(32))=(5/8) (1)⇒ Aire[(AOB)+(AOD)]=((68)/5)sin x=((17)/2) cm^2 donc : Reponse= e
$$\bigtriangleup\boldsymbol{\mathrm{OAB}}\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{2}} =\mathrm{41}−\mathrm{40cos}\:\boldsymbol{\mathrm{x}} \\ $$$$\:\:\boldsymbol{\mathrm{a}}=\sqrt{\mathrm{41}−\mathrm{40cos}\:\boldsymbol{\mathrm{x}}} \\ $$$$\begin{cases}{\mathrm{4sin}\:\boldsymbol{\mathrm{x}}=\:\:\boldsymbol{\mathrm{a}}\mathrm{sin}\:\boldsymbol{\mathrm{y}}}\\{\mathrm{4cos}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{a}}\mathrm{cos}\:\boldsymbol{\mathrm{y}}=\mathrm{5}}\end{cases} \\ $$$$\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OAB}}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{4}×\mathrm{5}\right)\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\mathrm{10sin}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OAD}}\right)\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{OC}}×\mathrm{3sin}\:\boldsymbol{\mathrm{x}} \\ $$$$=\:\frac{\mathrm{3}}{\mathrm{2}}×\frac{\mathrm{12}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\frac{\mathrm{18}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{Aire}}\left[\left(\boldsymbol{\mathrm{OAB}}\right)+\left(\boldsymbol{\mathrm{OCD}}\right)\right]=\frac{\mathrm{68}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$ \\ $$$$\boldsymbol{\mathrm{Calcul}}\:\boldsymbol{\mathrm{de}}\:\boldsymbol{\mathrm{x}} \\ $$$$\:\:\boldsymbol{\mathrm{h}}=\left(\mathrm{4}+\frac{\mathrm{12}}{\mathrm{5}}\right)\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\frac{\mathrm{32}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}} \\ $$$$\:\:\boldsymbol{\mathrm{Aire}}\:\boldsymbol{\mathrm{totale}}=\frac{\left(\mathrm{3}+\mathrm{5}\right)\boldsymbol{\mathrm{h}}}{\mathrm{2}}=\mathrm{4}\boldsymbol{\mathrm{h}} \\ $$$$=\boldsymbol{\mathrm{Aird}}\left(\boldsymbol{\mathrm{BCD}}\right)+\boldsymbol{\mathrm{ACD}}= \\ $$$$\boldsymbol{\mathrm{Aire}}\:=\mathrm{2}\boldsymbol{\mathrm{A}}\mathrm{ird}\:\left(\boldsymbol{\mathrm{OCD}}\right)+\boldsymbol{\mathrm{A}}\mathrm{2}+\boldsymbol{\mathrm{A}}\mathrm{3} \\ $$$$=\boldsymbol{\mathrm{Aird}}\:\boldsymbol{\mathrm{T}}\mathrm{itale}\:−\boldsymbol{\mathrm{OAB}}+\boldsymbol{\mathrm{OCD}} \\ $$$$ \\ $$$$\frac{\mathrm{3ACsin}\:\boldsymbol{\mathrm{x}}}{\mathrm{2}}+\frac{\mathrm{3}\boldsymbol{\mathrm{BD}}\mathrm{sin}\:\boldsymbol{\mathrm{y}}}{\mathrm{2}}= \\ $$$$\frac{\mathrm{128sin}\:\boldsymbol{\mathrm{x}}}{\mathrm{5}}−\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OAB}}\right)+\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OCD}}\right) \\ $$$$\Rightarrow\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OAB}}\right)−\boldsymbol{\mathrm{Aire}}\left(\boldsymbol{\mathrm{OCD}}\right)\:\:\:\:\left(\mathrm{2}\right) \\ $$$$=\frac{\mathrm{128sin}\:\boldsymbol{\mathrm{x}}}{\mathrm{5}}−\frac{\mathrm{3}}{\mathrm{2}}\left(\boldsymbol{\mathrm{AC}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{BD}}\mathrm{sin}\:\boldsymbol{\mathrm{y}}\right) \\ $$$$\frac{\boldsymbol{\mathrm{OD}}}{\boldsymbol{\mathrm{BD}}}=\frac{\mathrm{12}/\mathrm{5}}{\mathrm{4}+\mathrm{12}/\mathrm{5}}=\frac{\mathrm{12}}{\mathrm{32}}=\frac{\mathrm{3}}{\mathrm{8}}\:\:\:\boldsymbol{\mathrm{BD}}=\frac{\mathrm{8}}{\mathrm{3}}\boldsymbol{\mathrm{OD}} \\ $$$$\:\frac{\boldsymbol{\mathrm{OD}}}{\boldsymbol{\mathrm{OB}}}=\frac{\mathrm{3}}{\mathrm{5}}\:\:\:\:\boldsymbol{\mathrm{OD}}=\frac{\mathrm{3}}{\mathrm{5}}\boldsymbol{\mathrm{a}} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{AC}}=\mathrm{4}+\frac{\mathrm{12}}{\mathrm{5}}=\frac{\mathrm{32}}{\mathrm{5}}}\\{\boldsymbol{\mathrm{BD}}=\frac{\mathrm{8}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{5}}\boldsymbol{\mathrm{a}}=\frac{\mathrm{8}}{\mathrm{5}}\boldsymbol{\mathrm{a}}}\end{cases} \\ $$$$ \\ $$$$\:\:\:\:\frac{\boldsymbol{\mathrm{OAB}}}{\boldsymbol{\mathrm{A}}\mathrm{O}\boldsymbol{\mathrm{D}}}=\frac{\mathrm{25}}{\mathrm{9}}\:\:\:\:\:\frac{\boldsymbol{\mathrm{OAB}}+\boldsymbol{\mathrm{A}}\mathrm{O}\boldsymbol{\mathrm{D}}}{\boldsymbol{\mathrm{OAB}}}=\frac{\mathrm{34}}{\mathrm{25}}\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\: \\ $$$$\bigtriangleup\boldsymbol{\mathrm{ABC}}\:\:\:\boldsymbol{\mathrm{AB}}\mathrm{sin}\:\boldsymbol{\lambda}=\boldsymbol{\mathrm{AC}}\mathrm{sin}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{b}}\mathrm{sin}\:\boldsymbol{\lambda}=\frac{\mathrm{32}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{b}}=\sqrt{\mathrm{25}+\left(\frac{\mathrm{32}}{\mathrm{5}}\right)^{\mathrm{2}} −\mathrm{64cos}\:\boldsymbol{\mathrm{x}}} \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\frac{\mathrm{32}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\sqrt{\frac{\mathrm{1649}}{\mathrm{25}}−\mathrm{64cos}\:\boldsymbol{\mathrm{x}}} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\mathrm{32}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1600cos}\:\mathrm{x}+\mathrm{625}=\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{cos}\:\mathrm{x}=\frac{\mathrm{25}}{\mathrm{32}}\:\:\:\:\mathrm{sin}\:\mathrm{x}=\frac{\sqrt{\mathrm{399}}}{\mathrm{32}}\cong\frac{\mathrm{20}}{\mathrm{32}}=\frac{\mathrm{5}}{\mathrm{8}} \\ $$$$\:\:\:\:\:\left(\mathrm{1}\right)\Rightarrow \\ $$$$\:\:\boldsymbol{\mathrm{Aire}}\left[\left(\mathrm{AOB}\right)+\left(\boldsymbol{\mathrm{A}}\mathrm{OD}\right)\right]=\frac{\mathrm{68}}{\mathrm{5}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\frac{\mathrm{17}}{\mathrm{2}}\:\mathrm{cm}^{\mathrm{2}} \\ $$$$\:\:\boldsymbol{\mathrm{donc}}\::\:\boldsymbol{\mathrm{R}}\mathrm{eponse}=\:\boldsymbol{\mathrm{e}} \\ $$

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